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Collection  de 
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D 


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n 


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Coloured  covers/ 
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I      I    Covers  damaged/ 


Couverture  endommagie 


Covers  restored  and/or  laminated/ 
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obtenir  la  meilleure  image  possible. 


The  C( 
to  the 


Their 
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of  the 
filmin 


Origin 
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the  lai 
sion, 
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which 

Maps, 
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entire 
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right  c 
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methc 


10X 

14X 

18X 

22X 

26X 

30X 

y 

12X 

16X 

20X 

24X 

28X 

32X 

slaire 
IS  details 
iques  du 
nt  modifier 
xiger  une 
ie  filmage 


The  copy  filmed  here  has  been  reproduced  thanics 
to  the  generosity  of: 

National  Library  of  Canada 


The  images  appearing  here  are  the  best  quality 
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beginning  with  the  front  cover  and  ending  on 
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other  original  copies  are  filmed  beginning  on  the 
first  page  with  a  printed  or  illustrated  impres- 
sion, and  ending  on  the  last  page  with  a  printed 
or  illustrated  impression. 


d/ 
ludes 


L'exemplaire  filmA  fut  reproduit  grice  A  la 
g6n6rositd  de: 

Bibliothdque  nationale  du  Canada 


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filmage. 

Les  exemplaires  originaux  dont  la  couverture  en 
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par  Ie  premier  plat  et  en  terminant  soit  par  la 
derniire  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration,  soit  par  Ie  second 
plat,  selon  Ie  cas.  Tous  les  autres  exemplaires 
originaux  sont  film^s  en  commen^ant  par  la 
premid>re  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration  et  en  terminant  par 
la  dernidre  page  qui  comporte  une  telle 
empreinte. 


The  last  recorded  frame  on  each  microfiche 
shall  contain  the  symbol  -^(meaning  "CON- 
TINUED"), or  the  symbol  V  (meaning  "END"), 
whichever  applies. 


Un  des  symboles  suivants  apparaftra  sur  la 
dernidre  image  de  cheque  microfiche,  selon  Ie 
cas:  Ie  symbole  — ►  signifie  "A  SUIVRE  ",  Ie 
symbola  V  signifie  "FIN". 


aire 


Maps,  plates,  charts,  etc.,  may  be  filmed  at 
iiff'trerft  reduction  ratios.  Those  too  large  to  be 
entirely  included  in  one  exposure  are  filmed 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  top  to  bottom,  as  many  frames  as 
required.  The  following  diagrams  illustrate  the 
method: 


Les  cartes,  planches,  tableaux,  etc.,  peuvent  dtre 
filmAs  A  des  taux  de  rMuction  diff6rents. 
Lorsque  Ie  document  est  trop  grand  pour  dtre 
reproduit  en  un  seul  cliche,  il  est  film6  A  partir 
de  I'angle  sup6rieur  gauche,  de  gauche  d  droite, 
et  de  haut  en  bas,  en  prenant  Ie  nomb'e 
d'images  ndcessaire.  Les  diagrammes  suivants 
illustrent  la  mithode. 


by  errata 
ned  to 

ent 

une  pelure, 

Papon  d 


1 

2 

3 

32X 


1 

2 

3 

4 

5 

6 

s 


WENTWOr^TH'S 
SERIES    OF    MATHEMATICS. 


First  Steps  in  Number. 

Primary  Arithmetic. 

Grammar  School  Arithmetic     ^ 

High  School  Arithmetic.  ♦ 

Exercises  in  Arithmetic. 

Shorter  Course  in  Algebra. 

Elements  of  Algebra.  Complete  Algebra. 

College  Algebra.  Exercises  in  Algebra. 

Plane  Geometry. 

Plane  and  Solid  Geometry. 

Exercises  in  Geometry. 

PI.  and  Sol.  Geometry  and  PI.  Trigonometry. 

Plane  Trigonometry  and  Tables. 

Plane  and  Spherical  Trigonometry. 

Surveying.  ^    ^ 

PI.  and  Sph.  Trigonometry,  Surveying,  and  Tables. 

Trigonometry,  Surveying,  and  Navigation. 

Trigonometry  Formulas. 

Logarithmic  and  TrigonometricTables  (Seven). 

Log.  and  Trig.  Tables  (jDompme  Edition), 

Analytic  Geometry. 


Special  Terms  and  Oironlar  on  Application. 


1" 


ALGEBRAIC  ANALYSIS. 


SOLUTIONS    AND    EXERCISES 


ILLUSTRATING 


THE   FUNDAMENTAL  THEOREMS  AND  THE 

MOST  IMPORTANT   PROCESSES  OF 

PURE  ALGEBRA. 


BY 

G.  A.  WENTWORTII,  A.M., 

Professor  of  Mathematics  in  Phillips  Exeter  Acadext; 

J.  A.  McLELLAN,  LL.D., 

Inspector  of  Normal  Schools,  and  Conductor  of 
Teachers'  Institutes,  for  Ontario,  Canada; 

AND 

J.  C.  GLASHAN, 


Inspector  of  Public  Schools,  Ottawa,  Canad 


PART  I. 


BOSTON,  U.S.A.:        w 

PUBLISHED   BY  GINN  &  COMPAN 

1889. 


Vii'1 


'^^^- 


Entered,  according  to  Act  of  Congress,  in  the  year  1889,  by 

G.  A.  WENTWORTH, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 

All  Rights  Resbrtkd. 


Typoorapht  by  J.  S.  CusHiNO  &  Co.,  Boston,  U.S.A. 


Presswork  bt  Ginn  &  Co.,  Boston,  U.S.A. 


I 


I 


PREFACE. 


rpillp]  work  of  whicli  tliis  volume  forms  the  first  or  introductory 
|>art  is  intended  to  su|>j»ly  students  of  mathematics  witli  a  well- 
lilled  storehouse  of  solved  examples  and  unsolved  exercises  in  the 
application  of  the  fundamental  theorems  and  processes  of  ])ure  Alge- 
bra, and  to  exhibit  to  them  the  highest  and  most  important  results 
of  modern  algebraic  analysis.  It  may  be  used  to  follow  and  sup- 
plement the  ordinary  text-books,  or  it  may  be  employed  as  a 
guide-book  and  work  of  reference,  in  a  course  of  instruction  under 
a  teacher  of  mathematics. 

The  following  are  some  of  the  special  features  of  this  volume  : 

It  gives  a  large  number  of  solutions  in  illustration  of  the  best 
methods  of  algebraic  fesolution  and  reduction,  some  of  which  are  not 
found  in  any  text-booh. 

It  gives,  classified  under  proper  heads  and  preceded  by  type- 
solutions,  a  great  number  of  exercises,  many  of  them  illustrating 
methods  and  principles  which  are  generally  ignored  in  elementary 
Algebras  ;  and  it  presents  these  solutiims  and  exercises  in  such  a 
way  that  the  student  not  only  sees  how  algebraic  transformations 
are  effected,  but  also  perceives  how  to  form  for  himself  as  many 
additional  examples  us  he  may  desire. 

It  shows  the  student  how  simple  principles  with  which  he  is  quite 
familiar,  may  be  applied  to  the  solution  of  questions  which  he  has 
thought  beyond  the  reach  of  these  principles ;  and  gives  complete 
explanations  and  illustrations  of  important  topics  which  are  omitted 
or  are  barely  touched  upon  in  the  ordinary  books,  such  as  the  Prin- 


I 


IV 


PREFACE. 


ciple  of  Symmetry,  Theory  of  Divisors,  and  its  application  to  Fac- 
toring, and  Applijations  of  Horner's  Division. 

A  few  of  the  exercises  are  chiefly  -supplementary  to  those  proposed 
in  the  text-books,  but  the  intelligent  student  will  find  that  even 
these  examples  have  not  been  selected  in  an  aimless  fashion  ;  ho  will 
recognize  that  they  are  really  expressions  of  certain  laws  ;  they  are 
in  fact  proposed  with  a  view  to  lead  him  to  investigate  these  laws 
fo!  himself  as  soon  as  ho  has  sufficiently  advanced  in  his  course. 
Nos.  8,  9,  10,  and  11  of  Ex.  i  afford  instances  of  such  exercises. 

Others  of  the  questions  proposed  are  preparatory  or  interpretation 
exercises.  These  might  well  have  been  omitted  were  it  not  that  they 
are  generally  omitted  from  the  text-books  and  are  too  often  neglected 
by  teachers.  Practice  in  the  interpretation  of  a  new  negation,  and 
in  expression  by  means  of  it,  should  always  precede  its  u.^e  as  a  sym- 
bolism itself  subject  to  operations.  Nos.  23  to  36  of  Ex.  3,  and  nearly 
the  whole  of  Ex.  15,  may  serve  for  instances. 

By  far  the  greater  numbor  of  the  exercises  is  intended  for  prac- 
tice in  the  nethods  exhibited  in  the  solved  examples.  As  many  as 
possible  ( f  tii^se  have  been  selected  for  their  intrinsic  value.  They 
have  been  ga  ihered  from  the  works  of  the  great  masters  of  analysis, 
and  the  student  who  proceeds  to  the  higher  branches  of  mathematics 
will  meet  again  with  these  examples  and  exercises,  and  will  find 
his  pi  ogress  aided  by  his  familiarity  with  them,  and  will  not  have 
to  interrupt  his  advanced  studies  to  learn  theorems  and  processes 
properly  belonging  to  elementary  Algebra.  In  making  this  selec- 
tion, it  has  been  found  that  the  most  widely  useful  transformations 
are,  at  the  same  time,  those  that  best  exhibit  the  methods  of  reduc- 
tion here  explained,  so  that  they  have  hhus  a  double  advantage. 

The  present  volume  ends  with  an  extensive  collection  of  exercises 
in  Determinants.  These  present  under  new  forms  and  from  a  dif- 
ferent point  of  view  the  greater  number  of  the  theorems  proposed, 
and  many  of  the  general  results  obtained,  in  the  earlier  chapters,  and 
to  these  they  add  many  important  propositions  in  other  subjects  ;  as, 


) 


PREFACE. 


ion  to  Fac- 

je  proposed 
that  even 
)n  ;  ho  will 
1 ;  thev  are 
these  laws 
his  course, 
cises. 

erpretation 
t  that  they 
1  neglected 
Nation,  and 
5  as  a  sym- 
and  nearly 

for  prac- 

lS  many  as 

ue.     They 

f  analysis, 

athematics 

will  find 

not  have 

processes 

this  selec- 

"ormations 

of  reduc- 

itage. 

exercises 

rom  a  dif- 

proposed, 

pters,  and 

jjecth ,  its, 


I 


) 


for  example,  in  the  method  of  least  squares,  in  linear,  homographic, 
orthogonal,  and  homaloid  transformatioTis,  and  in  the  degeneracy 
and  the  tangency  of  quadrics. 

The  second  volume  will  treat  of  factorials  and  the  combinatory 
analysis;  finite  differences  and  derived  functions,  bfth  direct  and 
invt  rse,  of  explicit  functions  of  a  single  variable  ;  expansion,  sum- 
ma*ion,  reversion,  transformation,  and  interpolation  of  seriee  ;  the 
arithmetic,  h£iiuonic,  and  geometric  series  of  integral  orders,  includ- 
ing the  theta-functions ;  recurring  series  ;  binomial,  logarithmic,  and 
exponential  series  ;  hyperbolic  and  circular  functions  ;  trigonometric 
series,  direct  and  inverse  ;  Legendre's,  Bessel's,  Lamp's,  and  Heine's 
series  and  their  associated  functions  ;  double  series  ;  infinite  prod- 
ucts ;  continued  fractions  ;  indeterminate  equations;  theory  of  num- 
bers ;  inequalities;  maxima  and  minima;  binomial  equations  and 
cyclotomic  functions  ;  transformation  of  binary  forms  ;  theory  of  Ihe 
quintic  and  of  higher  equations;  theory  of  substitutions.  The  whole 
will  close  with  a  chapter  on  the  fundamental  postulates  and  the 
general  laws  of  algebra,  illustrated  by  examples  and  problems  in 
matrices,  polar  algebras,  and  ideal  arithmetic. 

In  this  second  part  of  the  work  the  authors  hope  to  be  able  to 
give  numerous  historical  notes  and  bibliographical  references  for 
the  use  of  students  who  desire  to  pursue  the  subject  further,  or 
to  consult  the  original  memoirs. 

A  companion  volume  to  the  present  is  in  course  of  preparation  for 
the  use  of  private  students  and  of  all  who  have  not  the  advantage 
of  instruction  by  a  specialist  in  mathematics.  The  companion  will 
contain  proofs  of  the  theorems  employed  and  solutions  of  the  exer- 
cises proposed  in  this  volume,  the  whole  accompanied  by  hints  on 
the  best  method  of  attacking  problems,  and  on  the  selection  of  pro- 
cesses for  their  reduction. 

Notwithstanding  that  the  utmost  care  has  been  taken  in  revis- 
ing the  proof-sheets,  there  doubtless  remain  many  errors  both  in 
the  examples  and  in  the  exercises.     The  authors  would  feel  grate- 


VI 


PREFACE. 


I'ul  to  teachers  and  studenta  for  notification  of  all  errors  which 
may  be  discovered,  and  also  for  suggestions  in  relation  to  the  im- 
provement of  the  work. 

Messrs.  J.  S.  Gushing  &  Co,  deserve  special  mention  for  their 
masterly  skill  in  overcoming  all  the  difficulties  in  the  typography 
of  this  work,  and  for  their  excellent  taste  and  judgment  exhibited 
in  the  beauty  and  elegance  of  these  pages. 

(}.   A.   WRNTWORTII. 
J.  A.    McLKLLAN. 
J.   C.   GLASHAN. 


Si 


Note.   It  is  due  Mr.  Hlashan  to  state  that  the  main  part  of  the 
work  on  this  Algebra  has  been  done  by  him. 


(i.   A.  Wentworth. 
J.   A.    McLellan. 


Pr 


Fa 


M: 


>r8  which 
)  the  im- 


for  their 
l^ography 
exhibited 

ORTII. 

N. 


rt  of  the 

RTH. 

N. 


CONTKNTB. 


-•o*- 


CHAPTER   I. 

Substitution,  Horner's  Division,  Etc. 

Numerical  and  Literal  Substitution     . 
Fundamental  Formulas  and  their  Applications 
Expansion  of  Binomials      .... 


PAQK 

1 

13 

26 


Horner's  Methods  of  Multiplication  and  Division,  and  their 
Applications 27 

CHAPTER   II. 


Principle  of  Symmetry,  Etc. 

The  Principle  of  Symmetry  and  its  Applications 
The  Theory  of  Divisors  and  its  Applications 


39 
49 


Factoe,™.  chapter   III. 

Direct  Application  of  the  Fundamental  Formulas 
Extended  Application  of  the  Formulas 

Factoring  by  Parts 

Application  of  the  Theory  of  Divisors 
Factoring  a  Polynome  by  Trial  Divisors     . 

CHAPTER   IV. 

Measures  and  Multiples,  Etc. 

Division,  Measures,  and  Multiples 

Fractions 

Ratios 

Complete  Squares,  Cubes,  Etc 


77 

88 

100 

105 

114 


130 
140 
155 
1(34 


vm 


CONTENTS. 


CHAPTrni   V. 

lilNEAK    EQUATIONS   OF    OnK    CNKNitWN    (irANTITV. 

PAUI 

Preliminary  K(iuationH 171 

Fractional  Equationn 173 

Application  of  Uatios 179 

KeHolution  by  Rejection  of  ('oiiHtant  FactorH       ,         .         .  183 
Higher   K(|uati<)nH  which    aro    Kenolvahlo   into    Rational 

Linear  FactorH 191 


CIlAl'TKR   VI. 

Simultaneous  Linear  Equations. 

{Systems  of  Equations  .... 
Apj)lication  of  Syinmetrv    . 
I'articular  Systems  of  Linear  Equations 


202 
207 
211 


CHAPTER   VII. 
Quadratic  Equations. 

Pure  Quadratics 226 

Quadratic  Etpiations  and  Equations  that  can  be  solved  as 

(Quadratics        . 231 

Simultaneous  Quadratic  Equations 238 


CHAPTER  VIII. 


Indices  and  Surds. 


Indices  and  Surds 263 

Complex  Quantities 273 

Surd  Equations 277 


CHAPTER  IX. 

Cubic  ..^nd  Quartic  Equations 

Cubic  Equations  .        .         .        . 

Quartic  Equations       .        .         .        . 


.    297 
.    305 


CONTENTS. 


U 


anal 


PAUI 

171 
173 
170 
183 

191 


Dktkhminants. 


CIUrTER  X. 


DofinitioriH  and  Notation     . 
Transformation  of  Detorniinantfl 
Evolution  of  DftterrninantH. 
Multiplication  of  DcteniiiuantH  . 
Applications  of  Dutcrniiuanta      . 


PAOB 

'MV2 
.'M2 
348 


202 
207 
211 


226 

231 
238 


263 
273 
277 


297 
305 


ALGEBRA. 


-•o*- 


•     }.  CHAPTEE  I.  —  SUBSTITUTION. 

Ex.  1. 

1.    If  a=l,  b  =  2,  c-^S,  d=r:4:,  x  =  ^,  3^  =  8,    find  the 
values  of  the  following  expressions  : 


a~{x~i/)~(b~c)  {d  -  a)    -  (y/  -  h)  (x  +  c) ; 

^  -  I/ly  -  (1/  ~  a)\d  +  c{b  -~  c)l]; 

(x  +  d){i/-\-b-\-c)  +  ix-d)(a~b-d) 
'h(^  +  d)(a  —  x  —  d); 

(d~xy-\-(c  +  yy- 

(a  -  b)  {c'  -~b'x)~  {c  -  d)  (b'  -  a'x) 
+  (d-b-c)(d'~-a')', 


d  —  a  .  c?  + 


+ 


4+b 


d-\-a     d~c        d~b 


2.    If  a  =  3,  ^>  =  — 4,  c=— 9,  and  2s  =  a+o  -f- c,   find 
the  values  of  the  following  expressions : 

s{s  —  a){8  —  b){s~c)\ 
B'-V{s-ay^{s~bY-^{,~cy; 
s'~(s-a)  (s~b)~  (s  ~  h)  {s  -  c)  -  (s-c)  (s  -  a); 
2(s  -  «)  (s  -  i)  (s  -.  c)  +  a(s  -  ^»)  (s  -  c) 
+  i («  —  6")  (s  —  a)  4-  c(s  —  a)  (s  —  6?/. 


ij; 


SUBSTITUTION. 


3. 


U  a~2,  b  --    -  3,  c~\,  x~  4|^,  iind  the  values  of  the 
following  expressions : 


a 


]l^     d?-\-h\     (a ~ by  .     {a -  bf 


a'-ab-j-b''     a^-^ir     21      3  4      j      2     ' 


4Z»^c'^-(a^-^"^-c7^ '  ia-b)  {b   -  e)  {c-a) 


4.    If  «  =  6,  5  =  5,  c  =  —  4,  r;?  =  —  3,  find  the  values  of  the 
following  expressions: 

a'-    -yyib'  +  ac)  .       c  +  V(c^'^  +  C) 
2a-^(b''-ac)'     c' +  2d{d^  ~- c'j 


8 


5.    1^  X —S,  1/ —  4:,  z  =  0,  find  the  values  of 

\Sx -  v(^-^  +  y^)r!2:.-  +  VC^"  +  /+  2'0! ; 

a^y  H-  y^  +  ^-^ ;    (^'  -  yy-'  +  (y  -  -  2)''-  *  +  (2  -  ^O'"' ) 


■*^ 

6.    Find  the  values  of     ^    ^-/^  ^    _.    ^         ^          ^        . 
when 

(1.)   x—l,     y  —  2,     2-3; 

(ii.)   x-2,     y-3,     2-4; 

' 

(ill.)   a; -3,     y-4,     2-5; 

(iv.)  a; -10,  y-11,   2-12.       •   '  : 

SUBSTITUTION. 


jes  of  the 


U-1 
-a) 


lea  of  the 


-2ac)U 


■')"• : 


n 


7.  Given  .r  =  3,  ?/  =  4,  z  =    -  5,  find  the  values  of 

(a:  +  y -f  .r)' -  3(.r -f  y  f  z)(.r?/ +  yz  +  20:)  ; 

^'(y  ~  2)  +  3/'(2  -  ^)  +  2'(^  -  y) ; 

(5a:  -  4 z)^  f  9(4a:  -  z)'^ -  (13a:  -  bzf ; 

(3a:  +  4y  +  5z)'+(4a:4-32/  +  12z7-(52,-  +  52/  +  132/. 

8.  li  s  =  a-\-b-{-c,  find  the  values  of 

(2s  -  a)'H  (2s  -  ^>)' -  (2s  +  c/ 
when 

(i.)  a  —  3,  6  =  4,  c  =  b\ 

(ii.)  a  =  21,  6  =  20,  c  =  29; 

(iii.)  a=119,  6=120,  ^=169; 

(iv.)  a  =  3,  6  =  —  4,  c  =  5  ; 

(v.)  a  =  5,  6  =  12,  c  =  -  13. 

9.  If  a  =  l,  6  =  3,  c=5,  c?=  7,  c  =  9, /=  11,  show  that 

±+1  +  1+1  + i  =  iA_iY 

ab     he     cd     de     ef     2\a     fj ' 


1  +  1 


1  +  1 


VI 


1 


aba     hcd     cde     dcf      A.\ab      ef) ' 


1,+  1 


J_^l/  1 1 


abed     bedc     edef      6\abe      defj  ' 

::.  a'  +  b'  +  c'  -ab-be  —  ca  =  6''+  c'-\-  d'-  be ~ cd~  db 
—  e^  +  d'^  -\-  (?  —  cd  —  do  ~  cc 
=  d'-^(?-\-r-de~  cf-fd. 


SUBSTITUTION. 


10.    If  rt--l,   b=-2,   c  =  '6,   (/--4,    6'-- 5,  /=6,  ^=7, 
show  that 


ab{a^b)' 

a^j^b'^-^c-'^  d'  =:  ^y^^^+f)  ; 

rtw  (a  +  b) 

a-  +  6^  +  .^  +  c^  ^  +  c'^  -  4M4-^  ; 

a6  (ct  -f-  0) 

ao  {a  +  b) 
a?  4-  ^.••'  -f  c-'»  -  (a  +  5  +  cf  ; 

a«  +  ^>-^  +  c^  +  c?''  =  (a  +  Z>  +  c  -f  (Z)' ; 
a^^b'-{-  c'  '\'d'  +  (^:=(a-}-b  +  c  +  d-i-ey; 
a^j^b'-\-e'-^d'  +  e?  -{-/'  -  (a  +  b  +  c-{-d+e-\-fy; 

a*i-b*  +  c'==^Ai£±Al(^l::iIl- 

bc(b  ~{-  c) 

a'  +  i'  +  e^  +  a>^  Md  +  i)  (cdc^  1)  . 

oc  (o  +  o) 

„*  +  J.  +  ,.  +  rf.  +  e' =  ^+/ Wzii)  ; 
a+b  +ci-d  i-e+f bc{b  +  c) 


C^^d'  =  c^;     c^-^-d"" -[-(?=/ 


SUBSTITUTION. 


6,  i7  =  7, 


+/-Vy; 


11.    Assume  any  numerical  values  for  x,  y,  and  2,  and  find 
the  values  of  the  following  expressions : 

{x-\-Vf-2{x-{-bf-{x-^^f-\-2{x-\-l\f 
+  (^+12)'-(a;+16)»; 

{x^-y^y^{2xyy-{x^-\-yJ', 

{x^-2>xyy-\-(Zx'y-fY-{x'-\-yy- 

{Zx''-\-^xy-\-yy-\-{4:x'-\-2xyy—(J)x^-\-^xy-\-yy', 

ix-yr-{-{y-zy^{z-xy-^{x-y){y-z){z-x). 


-1) 


) 


§  1.  If  a;  =  any  number  (as.  for  example,  3),  then 
x\=  xxx)  =  3a;;  ar'(=  xXx^)  =  ^x'\  x\=  xXa?)=^  2>x^-, 
etc.  Or,  3  =  a:;  3a;  =  a;';  ^s^  —  x^]  3a;*  =  a,-^;  etc.  Hence, 
problems  like  the  following  may  be  solved  like  ordinary 
arithmetical  problems  in  "Reduction  Descending." 


Examples. 
1.    Find  the  value  of  ar^  —  2a;  —  9,  when  x 

x'  —  2x-^ 
5 


=  5. 


bx 

—  2x 

3a; 
5 

Is 

-9 


6  Ans. 


6 


SUBSTITUTION, 


2.    Find  the  value  of  x^  -  j?  —  ix^ 


5  when  x  =  3. 


X'' 


■x 


>x  --  i) 


Pi 


8r/ 


_  A_ 

/);, 6a; 

_:±? 

>'•, 3  a; 

3 

^4 9 

-5 

?'4 4  ^ws. 

3.    Find  the  value  of  2a;''+12a;"'+6a;'— 12a;+10  when  a;  =  -5. 

Using  coefficients  only,  we  have 

2+12  +  6-12  +  10 
-5 

2h-  -  '  •     —10 

+  12 

n  .....+  2 

-  5 

p,.  .  .  .  .-10 

±  _6 
^2 —  4 

-  5 

^3 20 

-  12 

^3 8 

p, -40 

+  10 

r^ —  30  Ans. 


I 


leii  X  -—  o. 
-  3  a:  -  -  5 


3n  07  =  — 5. 


2  +  10 


SUBSTITUTION, 


§  2>  If  the  coefficients,  and  also  the  values  of  a:,  are  small 
numbers,  much  of  the  above  may  be  done  mentally,  and  the 
work  will  then  be  very  compact.  Thus,  performing  men- 
tally the  multiplications  and  additions  (or  subtractions)  of 
the  coefficients,  and  merely  recording  the  partial  reductions 
^*i,  ^'21  *'3»  and  the  result  r^,  the  last  example  will  appear  as 
follows : 

-5)2     +12    +6    -12     +10 

2 
_.4 

8 
-30 

§  3.  In  the  above  examples,  the  coefficients  are  "brought 
down  "  and  written  below  the  products  p^,  p.^,  ps,  pt,  and  are 
added  or  subtracted,  as  the  case  may  require,  to  get  the 
partial  reductions  7*1,  rj,  rs,  and  the  result  7\.  Instead  of 
thus  "bringing  down"  the  coefficients,  we  may  "carry  up" 
the  products  pi,  p^,  JO3,  p^,  writing  them  beneath  their  cor- 
responding coefficients,  and  thus  get  ri,  r^,  r^,  r^,  in  a  third 
(horizontal)   line.     Arranged  in  this  way,  Exam.  2  will 

appear 

1     -1     -4    -3     -5 

+3     +6     +6    +9 


1+2+2 
and  Exam.  3  will  appear 


+  3; 


2+12      +6-12+10 
-10    -10    +20    -40 


2      +2      -4      +8;  -30 

Comparing  these  arrangements  with  those  first  given 
(Exams.  2  and  3),  it  will  be  seen  that  they  are,  figura  for 
figure,  the  same,  except  that  the  multiplier  is  not  repeated. 


SUBSTITUTION. 


■I'; 


§  4.  When  there  are  J^everal  figures  in  ^^e  value  of  x, 
they  may  be  arranged  in  a  column,  and  each  figure  used 
separately,  as  in  common  multiplication.  "When  only  ap- 
proximate values  are  required,  '  contracted  multiplication  " 
m.iy  be  used. 

4.    Find  the  value  of  3:r''-iG0a,-*  +  344:i;'  +  700a;'^— 1910a; 
+ 1200,  given  c  =  51. 


1 

50 

3 

-  IGO 

3 

150 

+  344 

-7 
350 

+  700 

13 

-650 

-  1910 
37 

1850 

+  1200 

23 

-1150 

3 

7 

13 

+  37 

-23; 

+  27 

.*.  result  is  27. 

5.    Given  a;  =  1.183,  find  the  value  of  64a;*  —  144x  +  45 
correct  to  three  decimal  places. 


1 
1 
8 

o 


64 


0 
64 
6.4 
5.12 
0.192 


0 

75.712 
7.5712 
6.0570 
0.2271 


144 

89.5673 
8.9567 

-  7.1654 
0.2687 


45 
38.0419 

-  3.8042 

-  3.0434 
-0.1141 


64  75.712  89.5673   -38.0419 
.'.  result  is —0.004. 

Ex.  2. 

Find  the  value  of 

1.  a;*-lia;'-lla:'-13a7  +  ll,  for  a;  =  12. 

2.  a,-*  +  50a;'»  — 16a;''-16a;  — 61,  for  a;  =  -17. 

3.  2a;*  4- 249a;' -1253;=^ +  100,  for  a;  =  —  125. 

4.  2a;' -473a,-'^- 234a; -711,  for  a;  =  200. 

5.  a;'^-3a;'^-8,  fbra;  =  4. 


0.0036 


6 

7 

8] 

9J 

lOj 


e  value  of  x, 
1  figure  used 
hen  only  ap- 
iltiplication  " 


+  1200 

-23 

-1150 


+  27 


144  a: +  45 

f  45 

-  38.0419 

-  3.8042 

-  3.0434 
0.1141 

0.0036 


SUBSTITUTION. 


9 


6.  .r*  -  -  515a;*  -  3127 :t-*  +  525ar'  -  2090a;»  +  3156a; 

-15792,  for  a;  =  521. 

7.  2.r*+401a;*-199r''  +  399a;'-602a;  +  211, 

for  a;:-  — 201. 

8.  1000a;*  — 81a;,  for  a; -=0.1. 

9.  99a;*+117ar'-257a;"-325a;-50,  fora;  =  lf 

10.  5a,'*  +  497a;*  +  200 ar»  +  196a;'  -  218a;  -  2000, 

for  a;  =-99. 

11.  5a;*  -  620a;*  -  1030a;»  +  1045a;'  -  4120a;  +  9000, 

fora;  =  205. 

Calculate,  correct  to  three  places  of  decimals : 

12.  a;'+3a;'-13a;-38,fora;=3.58443,fora;=-3.77931, 

and  for  a;=  — 2.80512. 

13.  y*- 14y'-h  2/ +  38,  for  y  =  3.13131,  for  y=- 1.84813, 

and  for  y  =  — 3.28319.  ... 


Ex.  3. 

What  do  the  following  expressions  become  (i.)  when  x  —  a; 
(ii.)  when  x=  —  a? 

1.  ai*  — 4aar'  + 6a' a;'- 4a' a;  +  a*. 

2.  V(^-«^+a')-  3.    ^(a;'  +  2aa;  +  a'). 

4.  (x'-\-ax+ay-{p^~ax-\-ay. 

If  x~y  —  z  —  a,  find  the  value  of  the  following  expres- 
sions : 

5.  ix  —  y){y~z){z~x). 

6.  {x-^yy{y~\-z~a){x-\-z~a). 


10 


SUnSTITlITION. 


7 .  ^  (y  +  2)  {y'  4-  z'  -  x')  +  y  (2  -f  x)  (f  4-  x' 


-/) 


8. 


a: 


+ 


y 


+ 


y-fz      a: +  2      a;  +  ?/ 


Find  the  value  of 


X   ,   X 


9.      -f  7,  when  a:  = 


a^c 


tt 


a 


10. 


+ 


-[-b 


+ 


/I        .  .  ;.         X  .     f -,  whena:=~(a-Z»  +  c). 

a(6  — a:)      b{c  —  x)      a{x  —  c)  a 


11.  ?  +  ^,  when^==^I^. 
a     0  —  a  o{b-\-a) 

12.  (a  +  a:)(i  +  a;)-a(Z>  +  c)+a:*,  whenar^^- 
13.   hx-\-  cy  -\-  az,  when  a;  =  ^-|-c  —  a,  y  =  c-{-  a  —  h, 

z  =  a-\-b~  c. 


a 


-,  when  a:  =  —  a. 


j4^    a(l  +  ^>)  +  3a: 

a  (1  +  <^>')  —  hx     a  —  2  6a:' 

16.  (^-^)(a;  +  2r)  +  (r-a:)(p  +  ^),  when  a:  =  ^^i^^^. 

2q 

17.  a^(Z»-c)  +  Z»2(c-a)-f  c^(a_S),  whena-5  =  0. 

18.  {a^b'\rc){hc-{-ca^ah)-{a'\-b){b-\-c){c-\-a), 

when  a  =  —  5.  •. 

19.  (a  +  6  4- 1?)'-  (a'  +  6'  4-  <?').  when  a-\-b=.0, 

20.  (^'4-2/ 4- 2)*-(^4-2/)*-(y4-2/-(2 4-^)* 4-^*4-/4-2*, 

when  a;  4-  y  4-  2  =  0. 

21 .  a'{c~b'')-\-  P {a  -  c')  -\-  c' (b  -  u')  4-  abc {abc  -  1), 

when  i  —  a^  =  0. 


S 


2 

2 


SUHSTITITION. 


11 


y'O 


{a~b  +  c). 


a  —  b, 


b  =  0. 


22.    d' 


(i^M 


+  /> 


a" 


ft\5 


whon  a*  -f-  i**  — -  0. 


23.  ExpreRs  in  words  the  fact  that  (a  —  by"a?  —  2ab-{-b\ 

24.  Express  algebraically  the  fact  that  "the  sum  of  two 

numbers  multiplied  by  their  diffeTenco  is  otpial  to 
the  difference  of  the  squares  of  the  numbers." 

25.  The  area  of  the  walls  of  a  room  is  equal  to  the  height 

multiplied  by  twice  the  sum  of  the  length  and 
breadth.  What  are  tlie  areas  of  the  walls  in  the 
following  cases :  (i.)  length  I,  height  A,  breadth  b ; 
(ii.)  height  x,  length  b  feet  more  than  the  height, 
and  breadth  b  feet  less  than  the  height. 

26.  Express  in  words  the  statement  that 

(x  -{-  a)  (x -\- b)  —  a^  -\-  {a -{- b)  X -{■  ab. 

27.  Express  in  symbols  the  statement  that  "the  square  of 

the  sum  of  two  numbers  exceeds  the  sum  of  their 
squares  by  twice  their  product." 

28.  Express  in  words  the  algebraic  statement, 

(a;  +  y)»  =  ar*  +  y  +  3a:2/(a: -f  y). 

29.  Express  algebraically  the  fact  that  "the  cube  of  the 

difference  of  two  numbers  is  equal  to  the  difference 
of  the  cubes  of  the  numbers  diminished  by  three 
times  the  product  of  the  numbers  multiplied  by 
their  difference." 

30.  If  the  sum  of  the  cubes  of  two  numbers  be  divided  by 

the  sum  of  the  numbers,  the  quotient  is  equal  to  the 
square  of  their  difference  increased  by  their  product. 
Express  this  algebraically. 

31 .  Express  in  words  the  following  algebraic  statement : 

-^^  -  (a;  +  y)' -  a;3/. 


12 


SUBSTITUTION. 


32.  Tho  Hquare  on  the  diagonal  of  a  cube  is  oqual  to  three 

timeH  tho  square  on  the  odgo.  Exprt;H8  this  in  sym- 
bols, using  I  for  length  of  tho  edgo,  and  d  for  length 
of  the  diagonal. 

33.  Express  in  symbols  that  "  the  length  of  the  edge  of  the 

greatest  cube  that  can  be  cut  from  a  sphere  is  equal 
to  the  square  root  of  one-third  the  square  of  the 
diameter." 

34.  Express  in  symbols  that  any  "rectangle  is  half  the 

rectangle  contained  by  the  diagonals  of  the  squares 
upon  two  adjacent  sides." 

The  Rquare  on  the  diagonal  of  a  square  is  double  the  square 
on  a  side. 

35.  The  area  of  a  circle  is  equal  to  tt  times  the  square  of 

the  radius.  Express  this  in  symbols.  Also  express 
in  symbols  the  area  of  the  ring  between  two  con- 
centric circles. 

36.  The  volume  of  a  cylinder  is  equal  to  the  product  of  its 

height  into  the  area  of  the  base ;  that  of  a  cone  is 
one-third  of  this ;  and  that  of  a  sphere  is  two-thirds 
of  the  volume  of  the  circumscribing  cylinder.  Ex- 
press these  facts  in  symbols,  using  h  for  the  height 
of  the  cylinder,  and  r  for  the  radius  of  its  base. 


Ex.  4. 

Perform  the  additions  in  the  following  cases : 

1.  (h  —  a)x-\-  (c—h)y  and  {a -{•  h) x -\- {h -\-  c)y. 

2.  ax  —  hy^  {a—h)x—{a-\-h)y,  and  {a-\-h)x  —  {h 

3.  {y  —  z)a^^{z  —  x)ah-^{x-y)h'' 

and  {x  ~y)  a^  —  {z  —  y)  ab  —  (x  —  z)  b^. 


a)y. 


FUNDAMENTAL    FORMULAS. 


18 


il  to  three 
lis  in  Hym- 
for  length 

(Igo  of  the 
e  is  oqual 
re  of  the 

half  the 
e  squares 

the  square 

jquare  of 
0  express 
two  con- 

ict  of  its 
cone  is 

'^o-thirds 
Ex- 
height 


sr. 


ise. 


a)y. 


4.    ax  -f  hy  -}-  cz,  bx  -i-  ci/  -{•  az,  and  ex  -{-  ay  -{•  hz. 

6.    {a\rh):^-\-{h~\-c)y'-^{a-\-c)z\  {b-\-i')x'-\-{a-\-c)y' 
h  (a  -f  h)z\  (a  -f  c)x''  +  (a  -r  b)y''  +  {b  -f-  c)z\  and 
-(a-f-i-f-^)(^-f3/'  +  2> 

6.  .r(a-Z»)'-}-y(6-c)^4-2(^-a)',  y{a-by  rz{b    -cf 

-\-x{c  —  of,  and  2  («  —  /»)'  -f-  a;  (6  -  -  c)'  -f  y  (^  ~  «)'• 

7.  {a--b)x'-\-{b-c)f-^{c~a)z\  (b-c)x'' ■{■{c~a)f 

-^{a-b)  z\  and  {c  —  a)  a?  -\-  (a  -b)f-\-{b-  c)  z\ 

8.  {(i-\-b)x  ~{-{b -\- c)y—{c-\- (i)z,  {b  ■}- c)  z -\- {c -{- a)  x 

—  (a  -f-  b)y,  and  {a  -\-  c)y  '\-  {a -\- b)  z  -  -  {b  -{■  c)x. 

9.  a^-Zab-Wb\   2i»-}^>'  +  c?^   a6  - -J 6"  +  Z*\   and 

2ab-\b\ 

10.  aa:"-3ia:",  —  9  aa;*»  +  7  5a;*,  and  —  85a;'*-}- lOaa;*. 

11.  What  will(aa;— 5y-|-^2)-|-(5a;-|-c2/  — az)— (ca;-f-ay-|-52) 

become  when  x—y—z  =  1? 


Fundamental  Formulas  and  their  Application. 

By  Multiplication  we  obtain 

{x  -\-  r){x  -{-  s)  =  a?  -{•  (r  -\-  s)  X  -\-  rs  [A] 

{x-{-r){x  +  s){x-\-t) 

=  a^  -{-  (r  -\-  s  -{-  t)^?  ^  {rs  +  St  -\-  tr)x  -^  rst         [B] 

From  [A]  we  obtain  immediately 

{x±:yy  =  x',t2xy-^f                                        ;  [1] 

{x-\-y  +  zy  =  o^  +  2xy-^2xz-{-f-\-2yz-\-z'  [2J 

{-S^ay^^a'-^^^ab  [3] 

(xJ^y)(^x-y)=^a^~y^                                  '       '  [4] 
The  symbol  2  means  "  the  sum  of  all  such  terms  as." 


14 


FUNDAMENTAL    FORMULAS. 


From  [B]  wo  derive 

(x  rb  ^y  =  ^  ±  ^x^y  -f  2txif  ±1/  [5] 

=  ar'  ±  y''  ±.^xi/  {x  ±7/)  [GJ 
{x-\-,ji-zy  =  x'-^7/-\-z' 

-\-^xyz  [7] 

=  r'  +/  +  z»  +  3(:r  +  y)(2/  4-  z)(a;  +  0)  [8] 

=  :i^-\r'iy-\-z^-\-Z{x\y-{-z){xy-\-yz-^xz)  —  '6xyz    [9] 

(5a)''  =  :Sa'-f-35a^Z»  +  GSa/>t?  [10] 


Formula  [1].     Examples. 

1.  We  have  at  once  {pc  ■\- yf -\- {x  —  yY  =  2{2^ -\- y^)  and 

{x-\-yy  —  {x-yy^^xy. 

2.  {a  -\-h  -^  c  -^  dy  ■\-  (a  —  h  —  c  ■\-  dy   may   be    written 

[(a  -f  cZ)  +  (&  +  ^)?  +  [(a  +  c^)  -  ih  +  c.)P, 
which  (Exam.  1) 

=  2\{a-^dy^{h-\^cy\, 

similarly, 

{a  —  h^c-dy^{a^h~c-dy 

=  \{a~d)-{h-c)J^\{a-d)^{h-c)f 

.-.  {a-^h-\-c-\-dy-\-{a-h-c^dy^{a-h-\-c-dy 
-\-{a-\-h-c-dy 

\=<2.\{a-\-dy-^{h-\-cy-^{a--dy-\-(h-cy\ 

by  Exam.  1, 

=  4(a'^4-i'  +  e'  +  ^').  . 


FUNDAMENTAL   FORMULAS. 


15 


[5] 

;o_ 

W{x  +  y) 

[7] 

[8] 

-?>xyz    [9 

[10] 


4-  y")  and 
e    written 


c)J 


-c~dy 


I 


3.  ^im^Miy  (a  +  b-{- cf  —  2{a-\-h  \-c)c-\-c\ 

This  is  the  square  of  a  binomial  of  which  the  first  terra 
is  (a-\-h-{-c),  and  the  second  —c.     Hence  it  equals 

4.  Simplify  (a  +  hY-2 (a^  +  h^) {a  +  hf  +  2(«*  +  h% 

By  Exam.  1,  2(a*  +  b')  =  (a'  +  bj  +  (a'  -  bj. 

Hence,  the  given  expression  equals 

(a  -j-by-2  (a'  +  b')  (a  +  bf  +  (a^  +  bj  f  (a^  -  by 

-  [(a  +  ^')'  -  (a'  +  b'')Y  +  (a'  -  bj 
=  a*-{-2a'b''-]-b*  =  (a''-{-by. 

Simplify  :  Ex.  6. 

1.  (x+Si/y  +  (x-^fy;  (ia'  +  SbJ-(^a'-SbJ. 
Show  that : 

2.  (7??a:  +  7iy)'^  + (war  — my)''  =  (m^ -f-^0C'^'4-y0• 
3.    (mx  —  W2/)'*  —  (nx  —  my)'  =  (w'  —  n^)  (x^  —  y*). 

Simplify  : 

4.  [(a  +  35)^  +  2(a+3i)(a-i)-f  («-i)''](a-i)'^ 

5.  (.r  +  3)^  +  (a:  +  4)^  -  (:r  +  5)^ 

and  (Ja;'-2y7-a3/^  +  2a;0^ 

6.  (a  +  ^'  +  c)'  +  (i  +  ^)'-2(6  +  c)(a  +  ^.4-c). 
Show  that : 

7.  {ax-\-  byf  +  {ex  +  c?y)'  +  (ay  -  bx)"  +  {cy  -  dxf 

=^{a'^-b'Jrc''{-d'){x'-{-yy 

Simplify: 

8.  (a;-3y74-(3a;^-y)'-2(3a;»-y)(a:-3y'). 


! 


16 


FUNDAMENTAL   FORMULAS. 


9.    ix'  -\-xy-y'y  -{a?  -xy~f)\ 

(1  +  2a;  +  4a;7+ (1  -  2a;  +  4a;^)^ 

10.  If  a  -f  6  =  —  \c,  show  that 

(2a-5y  +  (2Z.-c)2  +  (2c-a)'+2(2a-5)(25-c) 
+  2(2^.-c)(2c-a)  +  2(2c-a)(2a-^.)  =  TVc^ 

Simplify: 

11.  2(a -  ^.)^ -{a-^hj-,  (a'  +  4ai  +  ^•y -  (a'  +  ^>2)l 

12.  (a  +  ^•)2-(5^-c)^  +  (c+^)^-(c^+«)^ 

13.    (^x  - yy  +  (iy -  zf  +  (U-  xyi-2(hx-y){U-x) 
+  2(iy-z)az-a;)  +  2(ia;-3/)(Jy~2). 

Show  that : 

(^~yy+(^-zy  +  (z-xy 


14. 


=  2(a;-y)(z-3/)  +  2(y-a;)(2:-a;)+2(z-2/)(0-a;). 
Simplify : 

15.  a  +  xy-2a+x')(l  +  xy-{-2(l  +  x*). 

16.  (^+y  +  0)'-(a;+3/-0)2-(y-f2;_a;)''_(2-f.:r-y)l 

17.  (^-2y+32)»+(30-2y)^  +  2(a;-2y+30)(2y-32). 
18. 

19. 


(^+yy  +  (x-yy~2(x~yy(x+yy.      / 


20.  (5a  +  3i)-^+16(3a  +  ^>y-(13a  +  557. 
Show  that : 

21.  (Sa-by-{-(Sb~cy-^{Sc-~ay-2(bSa)(Sb-c) 

+  2(Sb-c)(Sc-a)~2(a~Sc)(Sa-b) 
-4(a  +  5  +  c)^  =  0.   .. 

22.  If  z"  =  2xy,  show  that  (2a;'  —  y'^y  +  (z''  —  2y''y 

+  (a;'  -  2^7  -  2(2a;'  -  f){f  -  23/') 

+  2(a,-'  -  2z'){f  -  2f)  -  2(a;'  -  2s')(2a;'  -  f) 

=  (a;  +  y)\ 


FUNDAMENTAL    FOKMl  LAS. 


17 


\{2b-c) 


t\c\ 


I  _j_  j;j 


){lz-x) 


f){z-x). 


y-^z). 

). 


U-c) 


f) 


Simplify 

23.  (l-\-x-\-x^  +  x^y  +  (1  -  X  -  x'  f  x^y 

-^-ii-x  +  x'-xy  +  iii-x-x'-x^y. 

24.  (ax -\-  bi/y-2(a''x^ -^-I'l/^^ax-^-ht/y -\-2(a*x*-\-b'f/*). 

Formulas  [2]  and  [3].     Examples. 


1.    {l-2x-j-Sxy-=l~ix+    6x' 

+    ^x'-Ux' 


-j-9x* 


l-4a;+10a;'^-12^-='  +  9x-* 


2.  (ab  ~{- be -\- cay 

=  a'b^  4-  2ab'c  +  2a'bc  +  b'c'  +  2abc'  +  a'a' 
=  a'b^-{-b^c^  +  c''a^  +  2abc(a~{-b  +  c). 

3.  [(x  +  yy  +  x'  +  i/'Y 

=  (x  +  yy  +  2(a;  +  yyix"  +  y^)  +  x'  +  2.^^'  f  y' 

-=  (^  +  y)*  +  (^  +  y)'  [(^  +  yY  +  (^-^  -  yy] 

=  2(a;  +  y)*  +  (^' -  2/7  +  ^*  +  2^^y^  +  y* 
-2[(a:  +  y)*  +  a;*  +  /]. 

4.  i^x^ -]- xy -\- yy 

=^x''-\-2x'y-\-2x^y'-\-x^y''-\-2xf-^y' 
=  (a;  +  y)V  +  ^'y'  +  y'  (^  +  y)'. 

5.  In  Exam.  3,  subst'  ate  b  —  c  for  a;,  <?  — a  for  y,  and  con- 

sequently b—fjj  for  a;+3/;    then,  since  {b  —  cCy  ^=. 
(a— by,  Exam.  3  gives 

=  2[(a-J)'  +  (J-c)'  +  (e-a)']. 

6.  Making  the  same  substitutions  in  Exam.  4,  we  have 
(a'  +  5^  4-  c»-a5  ~  be  ~  cay 

=  (a-by(b-cy+{b-cy(c-ay-{-(c~ay(a~by 


18 


FUNDAMENTAL   FORMULAS. 


or,  multiplying  both  sides  ])y  4, 

=  4  (a  -  by  (b  -  cy  +  4  (b-cy  (c-«)'+4  (c~ay  (a -by. 
Hence,  from  Exam.  5, 

(a-by  +  (b-cy-{-(c-ay 

-  2(a-by(b~cy+2(b~cy(c-ayi-2(c-ay(a-by. 

Expand:  ^^*  ®- 

1.  (l~2x-j~Sx'-4afy;  (l~x  +  x'~-x'y. 

2.  (l~2x  +  2x'~Sar'-~x*y;  (l  +  Sx  +  Sx' -{- .rj. 

3.  (2a-b-~c'~iy;  (1 -.^-f  y  +  2)^;  (ix-yi-6zy. 

4.  (x^~x'y-\-xf-~ify;  {ax-\-bx^^cx^-\.dx'y. 

5 .  Show  that  (a^  +  ^.^  +  c'')(a;2  +  y^  +  2^)  -  (aa;  +  %  -j-  ^2)2 

^{ay-bxy^{cx~azy^{bz-~cAjy, 

6.  Show  that  («  +  ^')  a;  +  (^  +  e)  y  +  (c  +  «)  z  multiplied 

by  {a-h)x-\-{b-c)y^{G-a)z  is  equal  to  the 
difference  of  the  squares  of  two  trinomials. 

7.  Sliowthat(a-^.)(rz-c)  +  (5-(?)(^>-a)-f-((?-a)((?-^) 

~\\{a-  by  -\-{b-  cy  +  {c-  ay]  =  o. 

8.  Simplify  [a~(b-c)Yi-[b-(c-a)Yi-[c-(a~b)y. 

9.  Show  that  (a'+b'  -xj+(a,'+bi'-xy+2(aa,  +  bb,y 

=  (a'  +  a,'  -  xj  +  (b'  +  b,'  -  ^^)^  +  2(ab  +  a^J^l 

10.  Show  that  [(a~b)  (b-c)i-(b-c)  (c-a)  +  (c-a)  (a-b)] '' 

=  (a-by(b~cy+(b~-cy(c-ay+(c-ay(a-by. 

11.  SqvLB,Ye2a~^bx~-^cx-]~2dx. 

12.  If  a;  +  y -f  2  =  0,  show  that 

13.  Show  that  a" (b  +  c?)'^  +  b' (c  +  a)^  +  c'  (a  +  i)" 

+  2a^>c?(a  +  Z> -f  c)  =  2(a^»  +  Z>c  +  ca/. 


FUNDAMENTAL    FORMULAS. 


19 


^)'{a~-by 


)\a-hy 


• 

yi-czf 

ultiplied 
1  to  the 

a)(c—b) 

n~b)y. 

+hb,y 

■i-b)Y 
I -by. 


II'. 


§  5.  To  apply  formula  [4]  to  obtain  the  product  of  two 
factors  which  differ  only  in  the  signs  of  some  of  their  terms, 
group  together  all  the  terms  whose  signs  are  the  same  in 
one  factor  as  they  are  in  the  other,  and  then  form  into  a 
second  group  all  the  other  terms. 

1 .    Multiply  a-\-b  —  c-}-dhy  a  —  b  —  o—  d. 

Here  the  first  group  is  a—c,  the  second  b-\-d.      Hence, 
we  have 

[{a-c)^(b  +  d)][(a-c)-{b^d)-\ 
=  (^a-cf-(b-^rd)\ 

-  [(1  +  30;^  +  (3^  +  a:^)]  [(1 -h  3a:^)  -  (3.^  +  o:^)] 
-(1  + 3:1-7 -(3a.- 4-^)' 

=  X  —  O  X   ~p  O  X    —  X  . 

3.    Find  the  continued  product   of   a-{-b-\-  c,    b-{-  c  —  a, 
c-\-a  —  b,  and  a-\-b-c. 

The  first  pair  of  factors  gives  [(b  -\~c)-]-a]  [(b  -}-<?)  —  «] 


=  (b  +  cf 


a 


hij^2bc-\-c' 


a\ 


The  second  pair  gives  \a  —  {b  —  <?)]  [a  -\-  {b  —  <?)] 

=  a^~b''^'2.bc-c\ 
The  only  term  whose  sign  is  the  same  in  both  these 

results  is  ^bc\  hence,  grouping  the  other  terms,  we 

have 
i^bc-\- {h'  4-  c^  -  a^)]  [2 bo  -  ih'  +  c^  -  a^)] 

=  {2bcy-{b''  +  c^-a'y 

--=2a^b''-\-2b''c'-\-2c^d'-a'-b'-c\ 

4.    Show  that  {a?-\-ab-\-b''y -ceh' =  {d'-\-aby-\-{ab-\-hJ. 
The  expression  =  (a^  +  b')  {a'  -\-2ab-\-  b') 
=  (a''  +  b')(ai-by 
^d'ia  +  by  +  b'ia+by 
=  (d'  +  aby-\-(ab-\-by.     '      •• 


20 


FUNDAMENTAL    FORMULAS. 


Ex.  7. 

1.  {a'-\-2ah-{-h-'){a'~2,ah-\^b''). 

2.  {\x'^~xy-{-f){^x^-^fJrxy).         . 

3.  {a'-  ah  -f  2 h') {a} -{-ab\-2 b')  ;    (x'  +  4 xy) {x' ~ 4 xy), 

4.  [{x  +  y)x-y  (x  -  y)]  [(x  -y)x-y(y  -  x)]. 

5.  Simplify  (x+ZXx-S)  +  (x-}-4:)(x~i)~(x-j-b)(x-~6). 

6.  Simplify  (1  +  xy  +  (1  _  ^y  _  2(1  -  a;^. 

7.  (x'  +  yy~-(2xyy-(x'~yy, 

8.  (2a=^-3^.^  +  4c^)(2a'^-l-3^/^-4c^). 

9.  (2a-f-Z»-3^)(6  +  3c-2a);  (2a~5-36')(^>-3c  -2a). 

10.  (x'-\-y*)(x^  +  f)(x-i-y)(x-7j). 

11.  CrM-^'y  +  y^)(:r^-^y  +  ?/^)(a;*-:i-2y'^  +  7/*). 

12.  (a^-b-ab~l)(a-j-b-j-ab  j-1). 

13.  If  a*  =  ^»*H  d?*,  show  that 

(«'  +  ^'  +  c^)  (^'^  +  c'  -  «'0  (c^  +  a'  -  b')  {a}  +  b'  -  6'0 

=  4^*6'*. 

Simplify: 

14.  {x'-^f~^xy){x'  +  y-^J^lxy). 

15.  (^*-2r^+3a;^-2.r+l)(a:*  +  2:i:H3a;H2^  +  l). 

16.  Multiply  {2x  - y) a?-{x-^ y) ax -{- x^  hy  {2x -y) a^ 

-\-  (x  +  y)  ax  —  x^. 

Show  that : 

17.  (a^  J^b'^  +  c'  +  ab^hc-]- caf  -  (ab  +  bc  +  caf 

=--(a  +  b-\^cy(a'  +  b'  +  c'). 

18.  (a'-i-b'  +  e'  +  ab  +  bc  +  cay^(a'-\-a.b-i-ca-bcy 

=  [(a  +  b)(b  +  c)y  +  [(b  +  c)(c  +  a)]\ 

19.  4(a5  +  6?c?V-(a2  +  Z>2_^2_^2>)2 

=  (a+b-\-c-d)  (a-i-b-ci-d)  (c-{-d^a~b)  (c-\-d-a-\-b). 


FUNDAMENTAL    FORMULAS. 


21 


{x'''  —  4iXy). 
+  5)(:i;-5). 


-3c  -2a). 


-  b'  ~  c') 


2a; -fl). 


y) 


a' 


20.  rind  the  product  of :   x^  -\-  y"^  -\~  z^  —  2xy  -\-  2xz  -  2yz 

and  x"^  -{-y'  -{-z"^  —  2xy-~  2xz  -f-  2yz. 

21.  (x'  +  f  +  xy^2)(x'-xy^2-ry')(x'-y'). 

22.  (l-6a  +  9ft')(i  +  2a  +  3a'>). 

23.  [(m-}-n)-f  (p  +  <2')]  (w.  —  (^-f-^?  — n). 

24.  1  -f-  ^  +  ^'^j  ^"^  -f-  ^  —  1,  a;'*  —  a;  -j-  1,  and  l-{-x  —  x^. 

25.  (a-^>7(a  +  ^>7(a^  +  Z»7(a*  +  Z>y. 

26.  Show  that  (.i^^  +  ^-^y  +  I/'f  (-c'  -  ^y  +  y')'  -  (•^•'/)' 

Formula  A.     Examples. 

1.    Multiply  a;^  —  a;  -f-  5  by  .2;'^  —  x  -  7. 

Here  the  common  term  is  a;'^  —  ru ;  the  other  terms,  -i-  5 
and  —7.    Hence,  the  product  equals 

(x'-xy  +  (~7  +  ^)(x'-x)  +  (-7x5) 
=  (x'~xy-2(x^-x)-S5 


X 


2x'-x^i-2x-S5. 


2.    (x  —  a){x  —  Sa)(x-{-4:a)(x  +  (ja). 

Taking  the  first  and  third  factors  together,  and  the 
second  and  fourth,  we  have  the  product  equals 


{x''  -i^Sax  -  4:a'')(x'  -}-3ax  -18 


a 


2N 


{x-i-Saxy-{4:a''i-lSa'')(x^i-Sax)-72a\etc. 


-  bey 


-a 


-hby 


Ex.  8. 


Find  the  products  of : 

1.  {x'-i-2x  +  S)(x''  +  2x-4:)]  (x-y-{-Sz)(x-y+bz). 

2.  (a;+l)(a7+5)(a;+2)(a:4-4);  (3^-j-a-b){x'-{-2b-a). 

3.  (a^-3)(a^-l)(a^  +  5)(a^  +  7);  (x*i-x'-i-l)(x*'i-x'-~2). 


22 


FlTNnAMKNTvNL    FORMl'LAS. 


4.    [(.^  +  yy  -  4 .;//]  [(x  4-  y)'^  -I   5 .a/l 


'/ 


X 


3  . 


5.  (.r"-|-'''   r  7)(.7-"      r/,--9); 

6.  (/i.v;-|  //  I   8)(,u;-hy4  7). 

7.  (.r  +  a   -y)(.6-  1  ..  }-3y). 

8.  (.r''«-l^;"-  ^/)(.6-*'"-|-.7-"      />). 

9.  a;.-*-y/-'-|  2)(fr*-   /-  4). 

11.  .r-2-hV2,  .'•■  2-iV'^.  •'■    2    V-.  ■'•     2-V3- 

12.  (.6-  I-  a  -I-  />)  (.7;  +  />  --  c)  (.?;  -    (f,  ^1-  />)  (.y  -i-  /;  -}-  c). 

13.  (a  -}-  i  -1-  6')  ((c.  h  ^^  -I-  ^^)-l  (fi  -h  0  +  (^)  (/>  -h  c  +  r*f ) 

--  ((<-.  +  />> +  6' -I- </)^ 

14.  ^]io\w  ihai{2a'\-2b-'c)(2hi-2c     <f) 

+  (2 c f  2 a-  h)  (2 a  i-2 b-  0)  +  (2 6-f  2 c-r/)  (2 H-2 «  ■  h) 

Formulas  [5]  and  [(>].     Examples. 

1.  We  get  at  once 

(x  +  yf  4  (,t'  -  y)''  -  2  :^-  Cr'^  +  3  if)  ; 
(:^-j-7/y'--  (.1;      7/y'-2y(3..M-y^). 

2.  Simplify  (a-i-b-[-(f-S(a-\^b^~cyC'i-S(a  +  b-i-c)c'-c\ 

This   comes   under  formula  [5],  the   first  term  being 
a-{-b-\-  c;  the  second,  —c.     Hence,  the  expression  is 

[(a  +  b  +  ,^-ey--^^(a~^by. 

3.  Show  that  (x'  +  r?/  +  y'')'  +  (^I'y  -    x'  -  y'^)^ 

-6.ry(.r*+:fV^  +  yO-8.rV^ 
This   comes  under   formula   [G],  the  first  term  being 
(x"^  -\-  xy  4-  if),  and  the^ second  —  {3?  —  xy  +  y^)  ;   we 
have,  therefore, 

{{x"  +  a;y  +  f)  -  (.r'  -  2-y  +  y^)]  •'  --.  (2  xyY  -  8  :^;^  y\ 


FUNDAMKNTA  I,    F(  )IIMI;LAS. 


28 


-V3. 


2H-2ft-Z/) 


;rm  being 
3ression  is 


rm  being 
-f);    we 


Simplify:  Ex.  9. 

1.  (1  -  xy  -h  (1  -f  .r^)^' ;   (:r  +  :^y)^'      (..^  •   x/zf 

2.  («  -h  2/jy  -  -  (a  -  /;)••' ;  (Sa      hf  --  (3a  -^  12 A 


3.    (.'t-f-y-2)^  |-3(.f-f-7/-^y' 


|-2''-f^3(.^  f  y-  z)z' 


4.  (a  -  Z»)-''  -\~  (a  -f  i)^'  +  Ga(a'  -  Z,'^). 

5.  (.^•--y)•^  +  Ct•  I  yy  +  3(T--yy(:,+y)-.--3(//     x)(,'  \,/y 
6.(1  +  X  +  xj  -.  (1  _  .,. -f-  xy  -~  iSx{  1  -f-  X'  -I-  x'). 

7.  0^   -/>  -^)H(^>  +  c)M-3(Z>  +  60'^(a-Z,-_,;) 

8.  (3:^-4//-f52)=^-(52-4yy-f3(52  -4yy(af   -4y  1-5.^) 

-3(3.^;-4y-|- 5^)2(5^-^4//). 

9.  (1  + ,. -f  ^,7' +  3  (1  -  ^)  (2 -I  :r) -I- (1  -  -  ;..)-\ 
Show  that :  , 

10.  a{a-2hy-h{h  ^2a)'-^{a      h)(a\^b)\ 

1 1 .  a'  («« -  2  A^)^  -I-  Z,='  (2  a=^  -  -  Z,'')-"^  ^  (a^«  _  U^  («''  -( •  ^■•7\ 
Simplify  : 

12.  {x'^~[  ^  +  2/y+6(;.^  +  y^;(.r*  +  .;y-f  y)+(^^  __^_fL^2).^ 

Show  that: 

13.  ciJ'Wi-2hJ  +  h\2a'-[-hy^{^a'hJ 

Simplify: 

(ax  +  b^y  +  «••«  7/  +  /?/^:t-3  _-  3  abx>/  (ax  +  %). 
15.    What  will  d'  +  Z*'*  +  6"'  -  3 r^/^c  become  when 


14. 


a 


-i-b-\-c  =  0. 


16.    Find  the  value  oi  x' ~f  +  z'  +  Sx'y'z' vihen 


^'-^'  +  z'  =  0. 


l>4 


I'TNDAMKNTAI.    FORM  T  LAS. 


Formulas  [7],  [8],  ani>  |9|.     Kxamtles. 

1.  Simplily  (2;^;      3y)^  I  (4y  -  5.;)"  |  (3a;     y)»      ' 

-  3(2a;   -3y)(4y    -5^-)(3^t-    -y). 

By  [9]  tlii.s  JH  Rooii  to  1)0 

[{2x  -  37/)  -f  (4.V  -  5a,-)  h  (3a;      y/)f      (())«  ^  0. 

2.  Trove  tliat  (a   -  hf  \  (h  -    *;)•'  \- (a  -    af 

^'6{a-h){h~c){c~-a). 

Ill  [9]  subsiituto  (T  --/>  for  a;,  i  -  c  for  ?/,  and  r?-  a  for  z\ 
for  those  values  a;  -f-  y  -}"  2  —  0,  and  the  identity  ap- 
pears at  once. 

In  [8]  let  a;  =  i  -f-  c  —  a,  ?/  =  c  -f-  a  —  h,  z~  (t-\~h'-c\ 
and  therefore,  a;-}-y  =  2c,  y-|-z  =  2«,  z  +  a;  =  26; 
and  this  identity  at  once  appears. 

Ex.  10. 

1.  Cube  the  following  : 

l  —  x-i-x'^\a  —  b  —  c;  1  —  2a;  +  3a;*  —  4  a;\ 

Simplify : 

2.  (a;»  +  2a;-l)''  +  (2a;-l)(a;'^4-2a;-2)-(a.-H3a;'^-l)\ 

Prove  that : 

3.  xijz  +  (x -f - y)Oj  +  z)(z  +  x)  =  (x-\-y-\- z) {xy^yz -f zx)^ 

4.  {ax  ~  Injf  -f  c^l^  —  V"^  +  3  abxy  (ax  —  by) 

==(a'-b')l:t^i-7/). 

*  Note  that  the  right-hand  member  is  formed  from  the  left-hand  one  by 
changing  additions  into  multiplications,  and  multiplications  into  additions ; 
hence,  in  (a;  +  y  -f  2)  X  (a;  X  y  +  y  X  z  +  z  X  a;)  the  signs  -f-  and  x  may  be 
interchanged  throughout  without  altering  the  value  of  the  expression. 


■•  "-''■w*i*iti.\i»*«V(i-.'-ji.  t'vp  ii;^£-  jit^niis--*.  1 


FUNDAMENTAI.    FORMtlLAS. 


or, 


0. 


—  a  for  z; 
sntity  ap- 

li-b-cf 
I  -|-  /j  —  c  ; 


x'  -  ly. 


+  z^)- 


nd  one  by 
additions ; 
X  may  be 
sion. 


Simplify  : 

5.  {x-\~i/-\-zy-\-{x     2yf\{,j      2zf^-{z~2xf 

■\~^x     y-2z){y-~z-2x){z~x-^  ^ly). 

6.  {'Ix'-    ?>y'  \Az'f  [-{'ly'     ^z'  \Ax:'f 

7.  {2ax-~hyf-\-{2hy      czY  \  (2c3 -«./•)' 

-\-2i{2ax-\hy      cz)(^ll)y  |  cz      ax)(^lcz  |  ax     by). 

Prove : 

8.  (.rM-3.r^y-y'/|  [^xy{x~[-y)Y 

9.  9(r^  +  3/^  +  2'')--(a;  +  7/-fz)'    -(4.7;  |-4y  (-r)(.r--yj' 

4-(4y  +  4z  +  a;)(^y-0y  +  (42-f-4:i--|-y)(z-a;)^ 

10.  If  a;  -}-  y  +  2  —  0,  hIiow  tliat  x^  -\-  if  \  z*'      ?>xyz. 

11.  Ifa.-  =  2?/+32,  show  that  a,-='~82/'-27z''-18a:yz--=0. 
Show  that : 

1 2 .  (.r^ + ^-y + /)' + (''^'  -  ^y  ■ !-  y')'  -f -  82*  -  c^''  (.1* }-  ^.y  [-  y*) 

=  0,  if^'^-l  y^-h2'  =  0. 

Prove  that : 

13.  8  (a  4-  ^>  +  cf  -  (a  +  Z»)'  -  (b  +  t?)""'  -  (c?  +  ^0' 

=  S(2a'j-  b  -^  c)(a  -\-2b  -^  c)(a  -{-b  -{-  2c). 

Prove  the  following : 

14.  {ax  —  byy  -f-  b^ y^  =  «^ ^^  +  3  abxy  (by  —  ax). 

15.  a'-{-b'-\-(^-Sabc 

■■=  i  [(«  -  by  f  (6  -  cy  +  (c  -  a)'^]  (a  +  Z»  +  c). 

+  (c  +  a-^»)(«  +  ^-^)] 

=  (a-\~b —c)(b-}- c— a)(c-{-a  —  b)-{-Sabc. 

17.    (a  +  i  +  c)'-3[«(^-c)H^(^-«)'+^(^«-^)'] 
=  a=*  +  iHc'  +  24aZ>c. 


Jli 


Kf  NDAMKNTAI.    FOUMriiAS. 


18.     (II    i    />    I    7r)(w        /,y    I    (/>    t    r    (    T^/)(A       <•)•' 

iCJfr  i  />      f)(l2A  I  r      (Oilir  |  a      />). 


20.    ll'   ./•'(//  !  :) 


It 


A'j/z      (i/tr,  show  that 


I'iXI'ANSlON    «»K    l')lNi)MlALS. 


\V<'  h:i\'o.  from  lonmiln  I  '> 


{a  \-/>y 


<r 


I   ;^r7> 


.X 


^//^   I   // 


iuul(il>l)iiig  hy  ff  i  A,  \V(M)l)tiiiii 

((fc  f  A)*    -:(^*  }    \fi'/>  I  (WrA-  I  4r/A'  |  A* ; 

inulliplyiiig  (liis  by  <f  \  A,  we  ohtiiiii 

(a.  I  -  A)-'  -  a'  h  5  a*  A  H  10  tc"'  A'  |  10  a'  A''  -h  5  r/A^  |  A\ 

From  Uu'se  ox{nn[>h»s  wo  (h'rivo  th»^  foUowiiiif  law  for  lli<' 
loniiation  of  tlio  tonus  in  the  expansion  of  n  \  A  to  iiny 
n  >[uiro<l  }»owor: 

T.  Tho  ('X])oncM»t  of  (r,  in  iho  Jirsf,  term,  is  that  of  tho  given 
powiM',  and  <J('cra(f!cs  by  unity  in  each  succeeding  term;  the 
exponent  of  A  begins  with  unity  in  the  second  term,  and 
incrcdscs  by  miity  in  eacli  succeeding  term. 

11.  riie  coelhcient  of  the  first  term  is  unity,  and  the  co- 
eilicient  of  any  other  term  is  found  by  multiplying  the 
ooetlicient  of  the  preceding  term  l)y  the  exponent  of  a  in 
that  term,  and  dividing  the  product  by  the  number  of  that 
preceding  term. 

It  will  be  observed  that  the  coefficients  equally  distant 
from  the  extremes  of  the  ex])aiision  are  etpial. 


)  \iih(' 


w  Tor  tln' 
/>  to  any 


10  ijjivcn 


1 

M-m  ;  ilio 

urin,  and 

tlic  co- 
pji,uj  the 
i)f  a  in 
of  that 

distant 


MI^F-TFPI.H'ATIMN    AND    I»IVrs|nV. 


*>7 


Ex.  11. 

2.    What  will   lt<'  tli»'  law  ol'  .v/<///.s',  if      ij  \n\  writ 


f.,r  y 


in|l|v 

:j.  iv\|):ind  (/^    i>yA<f-    'ihy ,  {'ii>    <i)\ 

4.  Kxpand  (1    !    ///)";  ('///    !    If;  C2vt   |    1)". 

5.  What  in  tlw  «'o<'lli<'i('nl  of  i1h>  I'ouith  tt'i-ui  in  {^u.      />f'! 

7.  In  tlic  ('Xi>ansi(»n  <>{'(<(.      hf'  the  tliinl  ttTni  iM  Gnr/."7/; 

find  the  (iUli  and  nixtli  t(>t'nis. 

8.  Sliow  th.'it  (.r  I  ?/)•'      .r'-  -;./'       n./y/fr  j  y^Cr''  !  .>■>/  \  /). 

9.  I'^iom  [«]  show  that  2[(//       A/'   i   (A       r/'  |    (r      nf] 

=  5 (a       Aj(A       r){r       ,,)[(„.■     hf  \   (A       r/ 

lIoRNHIl's  AIkTIIoDS  i)V  Mr  LTU'LK'ATK  >.N   AND  DIVISION. 

Examples. 

1.    Fiiul  tlio  prodnct  of  kj'^-\l.ir  \  m.v  \' n  and  a.r  |  A./;-|-r. 

Write  tliu  niultipiioi-  in  a  column  to  tlio  left  of  the  mul- 
tiplicand, placing  each  term  in  thii  same  liorizontal 
lino  witli  the  jiartial  product  it  gives: 

/•.r*     \lr?      -\-mx        -\n  (^ 


ax 

-Ihx 


alJuf*   -[-alx*   -\-a7nx^    -\(tn.r^  ......  pi 

-\U\v'  ~\hW       -\l>nix'-^bnx  p., 

-f-("H-''       -\  rJ.i:^    {-cinx-^cn y?;, 

akv'-Y{al-'\-hh)  x^~\-{cmi-]-hl-\-ch)  jc^-\-{un'\-hn^cl)  x' 

-\  (b7i-\~cm)x-]-cn P 


28 


MULTIPLICATION   AND   DIVISION. 


§  6.  The  above  example  has  been  given  in  full,  the  pow- 
ers of  X  being  inserted ;  in  the  following  example  detached 
coefficients  are  used.  It  is  evident  that,  if  the  coefficient 
of  the  first  term  of  the  multiplier  be  unity,  the  coefficients 
of  the  multiplicand  will  be  the  same  as  those  of  the  first 
partial  product,  and  may  be  used  for  them,  thus  saving  the 
repetition  of  a  line. 

2.    Multiply  ?>x'  —  2x^  —  2x  +  Z  by  x''-{-^x  -  2. 

1 

+  3 

-2 


3 

-2 

+ 

0 

-2 

+  3 

+  9 

— 

6 

+  0 

-6 

+   0 

G 

+  4 

0 

+   4 

-6 

2>x 

'-{-Ix' 

I2x 

'-{-2:^ 

-3a; 

^  +  13^- 

-6 

3.    Find  the  product  of  (a;  —  3)  (:i' +  4)  (a; -- 2) (:c  -  5). 


+  4 

1    -  3 

+  4         12 

1+1         12 

2          2+24 

-5 

11         14+24 
5     +5     +70    - 

-120 

x'      (jx"       9a;^  +  94^ 

-120 

4.    Multiply  a;'  — 4a;2  +  2a;-3by  2^*  — 3. 

1     -4     +2     -3 

{a?Xoi?  =  x^) 
0 

0         0 
+  12-6+9 


2 

2 

8 

+  4 

0 

0 

0 

0 

0 

0 

0 

0 

3 

8 

2x^  —  8a;^  +  4a;*  -  S)x^  +  12a,'''  -  6a,-  +  9 
In  this  example  the  missing  terms  of  the  multiplier  are 


MULTirLICATION    AND    DIVISION. 


29 


U,  the  pow- 

le  detached 

3  coefficient 

coefficients 

of  the  first 

saving  the 


^ 


-6 
-6 


5). 


<  oi^  =  x^) 


) 


plier  are 


su2iplied  by  zeros ;  but,  instead  of  writing  the  zeros  as  in 
the  example,  we  may,  as  in  ordinary  arithmetical  multipli- 
cation, "skip  a  line"  for  every  missing  term. 


5 .    Multiply  a;*  -  2  x"  +  1  by  x'  -  a;'  +  3. 


1 
-1 

+  3 


1+0-2+0+1 

„l-.0  +  2-0-l 

+  3  +  0-6  + OH- 3 


{x^  X  x^  -^  a^) 


3^.fi    j^.(Sx'     -7x^     +3 


Find  the  value  of 
{x  -[-  2)  (x  +  3)  (a:  +  4)  (x  +  5)  -  9  (x+  2)  (2;+ 3)  {x-{-  4) 
+  3(:c  + 2)(^+ 3)  +  77  (:r  +  2)  -  85. 


1  +5 
9 

-16 
+   3 

-   39 

+  77 

+  4 

1  -4 

+  4 

+  3 

1  +0 

+  3 

13 

+   0 

+  2 

i  +3 

+  2 

13 

+    G 

+  38 
-26 

+  76 
85 

\  x*  +  5a^-    7a;'^  +  12a; 


9 


7.    Find  the  coefficient  of  x*  in  the  product  of 
X*'  —  aa^ -^hx^  ~  cx-\-  d  and  x^  '\-px-^  q, 

1     —a    +6     —  c    -{-  d 
—  ap 

+  (i  — ap  +  5') 


+^ 


30 


MULTirLICATlON    AND    DIVISION. 


Ex.  12. 

Find  tlio  produrt  of: 

1 .  ( 1   !  - .?;  -f  .r'^  - 1  -  x^  -L  x')  ( 1  -  .r'  H-  .r''  -  x''  -f  .r"  -  x''  -|-  x'') . 

2.  (1  4 -  ^^'0  (1  -  x'  -I-  .r«)  (1  -f  X  +  .T^  +  x'  +  .T^). 

3.  (^-r))(.7H-G)(.r--7)(.r  f8);   (2.r5-:i'^+ l)(a:*-.r  +  2). 

5.    (G./-  -  :i'-'  4-  2.T*  ~  2.r'  +  2^:^  +  l^-'^  +  G)  (3.9;2+4,r  + 1). 

Obtain  the  coefficients  of  x*  and  lower  powers  in 
6 •     (1+  ^--^  —  -g- ^  ^-" T 6^" "~  TT8 ''^'  )(-•-■"  2"-^ ~ "g"^  ~ T^^  ~ TTS^  ) • 

7.  Multiply  2:1-^ -  x^  +  3:r - 4  by  Sc(/' ~2x''-x~l. 
Simplify  the  following : 

8.  (.i;-hl)(a;  +  2)Cr  +  3)  +  3(:r+l)(a,--|-2)--10(.T-hl)f9. 

9.  x(x  +  1)  (x  +  2)  (x  +3)      3:^;(.r  +  1)  (x  |-  2) 

^~^2x(x+l)-\-2x. 

10.  a;  Of    - 1)  (^^  -  2)  (x  -  3)  +  ^x(x  -  1)  (.r     2) 

-±v(x^-l)-^2x. 

11.  (.i;  -  ])(:u  H- 1)  (^--l-  3)(:i--f-  5)   -  14(2-  -  l)(:i;  +  1)  +  1. 

12.  Given  that  the  sum  of  the  four  following  factors  is  - 1, 

find  (i.)  the  product  of  the  first  pair  ;  (ii.)  the  product 
of  the  second  pair  ;  and  (iii.)  the  product  of  the  sum 
of  the  first  pair  by  the  sum  of  the  second  pair  : 
(i.)  X  -\-x'-\  x''-\-x'\ 

(ii.)  :7r -f  .^«  H^  a;«  +^^ 

(iii.)  a^\~a^i-x'''}-x'\ 

(iv.)  :if '\^  ii" -\' x"' -\- .v"'. 


13.    Given  that  the  sum  of  the  three  following  factors  is 


e<|ual  to      1,  find  their  product: 

(i.)  X  ■]-  x'  -\-x^  -|-^"'"", 
(ii.)  .r  +  a;^  +  :i;'°  +  .;", 
(iii.)  x'\-a^-\-x'  +  .r^ 


MITLTIPLICATTON    AND    DTVIRTON. 


31 


in 
^—  1. 

(^•+1)4-9. 

) 


t'  +  l)+l. 

tors  is  - 1, 
le  product 
;)f  the  sum 
[)air : 


factors  is 


§  7.  Were  it  required  to  divide  the  product  P  in  the  first 
of  the  .'il)ove  examples  by  ax^  -f-  hx  +  c,  it  is  evichnit  that 
couhl  we  find  and  subtract  from  P  the  partial  products  ^)._,. 
2h  (or,  what  would  give  the  same  result,  could  we  add  then'i 
with  the  sign  of  each  term  changed),  there  would  remain 
the  2')artial  product  ^i,  which,  divided  by  the  monomial  «;^•^ 
would  give  the  quotient  Q.  This  is  what  Horner's  method 
does,  the  change  of  sign  being  secured  by  changing  the 
signs  of  h  and  c,  which  are  factors  in  each  term  of  ^?2,i>':), 
respectively. 


1. 

—c 


rt.l" 


a^,T''-f-(rtZ-t-&^);c*+(am4-&?+c/;)r"'+(an+6m-f  cZ)x"'^+(?>w4-cw)a;+cn..P 
~hkx*  —his?  —hmx^         —hnx 

—ckx?  —clx^ 


—cmx 


■  ■  ■  .P2 


akx^ 


Valx^ 


->t-amx^ 


-\-anx^ .j[>i 


kx^         +Zx2 


^mx 


Vn 


Q 


The  dividend  and  divisor  are  arranged  as  in  the  exam- 
ple, the  sign  of  every  term  in  the  divisor,  except  the  first, 
being  changed  in  order  to  turn  the  subtractions  into  addi- 
tions. The  first  term  of  the  dividend  (akx^)  is  brought 
down  into  the  line  of pi ;  dividing  this  by  (fv\  the  first  term 
of  the  divisor,  we  get  Lv^,  the  first  tcrvi  of  the  quotient. 
Multiplying  this  term  kof  by  —hx  and  —n,  respectively, 
and  writing  the  products  in  the  proper  columns  and  rows, 
makes  all  ready  to  give  the  second  term  of  ^?,,  which  is  got 
by  simply  adding  up  the  second  column  of  the  work,  giving 
alx^.  Dividing  this  second  term  of  p,  })y  ax^  gives  Ix^,  the 
second  term  of  the  qxLoticnt.  Multiply  /.r  Ijy  -  hx  and  •  r', 
respectively,  and  proceed  in  the  same  way  as  was  done  in 
getting  the  second  term  of  the  quotient,  and  the  third  will 
be  obtained.  Repeating  the  steps,  the  conqdote  quotient 
and  the  remainder  will  finally  be  obtained. 

Should  the  coefficient  of  the  first  term  of  the  divisor  be 
unity,  the  coefficients  of  tlie  line   Q  will  be  the  same  as 


32 


MULTIPLICATION   AND   DIVISION. 


those  of  ^1,  and  the  line  Q  need  not  be  written  down,  since 
one  line  does  for  both. 

2.    Divide  3:r«+7a;^-12a:*+2r'-3.^'+  13a:-G  by  .r^+3.r-2. 

3     +7     -12  +  2    -3  +  13-G       {jf'-^x'^x*) 
-9     +    6-0    +6-   9 

4-    6-4    +0-   4  +  6 


^3 

+  2 


3^*-2ar'+    0-2a;4-3 

Compare  this  example  with  the  second  example  of  Horn- 
er's Multiplication,  performing  a  step  in  multiplication, 
then  the  corresponding  step  in  division  ;  then  another  step 
in  multiplication,  and  the  second  (corresponding)  step  in 
division  ;  and  so  on. 

3.    Divide  x'-^x^^^x'-^-l^ot^ -1x^12  hj :^~2,x^-{-2>x-\. 

1-3-1-0  -  4    +18  +  0    -  7    +12 

+  3        +3  +  0  -9    -36-27 

-3  -3  -0    +    9  +  36+27     {x'-^x^^x') 

+  1  +1+0-3-12-9 


x'-\-{)-2>x'~l2x-    9;    6a;''+8a;+3 

The  quotient  is  therefore  a;*  — 3 a;'^— 12a;— 9,  and  the 
remainder  6 a;^  +  Bar  +  3. 

4.    Divide  x^ -2,x' -bx^'\-2x''-\-ba?-{-^x''-{-l  by  a;^  +  2a;-l. 

The  zero  coefficient  in  the  divisor  may  be  inserted,  or  it 
may  be  omitted  and  allowance  made  for  it  in  the  2a;-line. 
See  Exams.  4  and  5  in  Multiplication. 

1-3+0-5+2+5+4+0+1 
-2+6+4-4-6+2 

1-3-2+2+3-1 


-2 
+  1 


l__3__2  +  2+3-l;    0  +  5  +  0 

(a;^  -^  a;^  =  oc").     The  quotient  is  therefore  c(^  —  3a:*  —  2x^ 
+  2a;'^  +  3a;  —  1,  and  the  remainder  5a;. 


MULTIPLICATION   AND    DIVISION. 


33 


I  down,  since 
.7.^24-3^-2. 


pie  of  Horn- 
ultiplication, 
another  step 
ing)  step  in 

-12 
9 


9,  and  the 

x''{-2x-l. 

serted,  or  it 
le  2a;-line. 


-3:r*-2 


X 


5.    Divide  lOx""  ~  llx^  -  Sx'  -j-  20a;'  ■[-  lOx'  -\-  2 
hy5a^-Sx^-\-2x-2. 

Arranging  as  in  the  ordinary  method,  we  have 


10- 

-11      3  +  20 

+  10+    0+    2 

+  3 

6      3        6 

+  12 

2 

4+   2 

+   4-8 

+  ^ 

+    4 

-2-4+8 

5 

2- 

-12+4 

24  -  12  +  10 

Quotient  =  2^*  —  a:^  —  2.^  +  4  + 


24a;'^-12a;  +  ]0 
bx^-'Sx'-\~2x~-2 


"We  first  draw  a  vertical  line  with  as  many  vertical 
columns  to  the  right  as  are  less  by  unity  than  the  number 
of  terms  in  the  divisor.  This  will  mark  the  point  at  which 
the  remainder  begins  to  be  formed.     We  then  divide  10  by 

5,  and  thus  obtain  the  first  coefficient  of  the  dividend.  We 
next  multiply  the  remaining  terms  of  the  divisor  by  the 
2  thus  obtained.  Adding  the  second  vertical  column  and 
dividing  by  5,  we  obtain  —1 ;  we  multiply  by  the  —1,  add 
the  next  column,  and  divide  the  sum  by  5,  and  so  on  for 
the  others. 

This  method  is  not,  however,  always  convenient.  If  the 
first  term  of  the  dividend  be  not  divisible  by  the  first  term 
of  the  divisor,  the  work  would  be  embarrassed  with  frac- 
tions.   We  may  then  proceed  as  in  the  following  examples  : 

6.  Diyides^-Sx'i-a^  +  Saf-x  +  Shj  2a^-i-x'-Sx-i~l. 

Let  2  a;  =  2/,  or  ^=^- 

2 

Substitute  ;^  for  x  in  the  dividend  and  divisor,  and  we 
have 


34 


MULTIPLICATlOxN    AND    DIVISION, 


y 


2' 


9^ 


2^       2 


_?/-2x3?/+2^?/+2^x3?/--2^y+2'x3 


.  7/ I-;/ -2X3?/ --I- 2-^ 

2'^ 
_^  ?/-  6  //*  f  4//+  24/- 16//  +  96 


-y'+;/-67/-}-4...4 


Dividing  ?/''-6?/*+4y"'+24y''-16y +  96  by  ?/-'M-y'-6y+4, 
l)y  the  ordinary  method,  and  the  quotient  by  2'\  we 
have 

?/-7y-f-17      1     39//-114y-28 
2=^  2'^'    y«4_y^_Gy  +  4" 

Substituting  for  y  its  value  2^,  and  simplifying,  we  get 

39.^•^-5'7a;-7 


7a.-      17      1 
"4        8"'   8 


B 


By  comparing  the  dividend  of  A  with  the  original  ques- 
tion, we  find  that  we  have  multiplied  the  successive  coeffi- 
cients of  the  dividend  by  2",  2\  2^,  etc.,  and  we  have  mul- 
tiplied the  successive  coefl[icients  of  the  divisor,  oinitiing  the 
first  term.,  by  the  same  numbers.  Dividing  then  by  Horn- 
er's division,  we  get  the  coefficients  1,  — 7,  17 ;  and  for 
coefficients  of  remainder,  —39,  114,  and  28.  The  first  three 
of  these  divided  by  2,  2\  2^,  are  the  coefficients  of  x^,  etc. ; 
and  —39,  etc.,  are  divided  by  1,  2,  2\  Hence,  t^  3  work 
will  stand  as  follows : 

^^5_3^.4_f_^3_|_    3^.2 _      ^j^    ^~2x'-\-x^-?>x-\-l 
12         4         8  16      32  12       4 


1       6     -f  4 

f  24 

16  +  96 

._1 

1     +7 

-17 

+  6 

+  6 

-42 

+  102 

4 

-   4 

+    28      68 

1-7     +17 

-39 

+  114  +  28 

1  -6    +4 


y 


-+i 


X3 


-6y+4...4 

-f//-6y+4, 
it  by  2'',  we 


ing,  we  get 
.  .  .  .  .  B 

iginal  ques- 
sive  coefR- 
have  mul- 
7nitting  the 
by  Horn- 
;  and  for 
first  three 
of  x^,  etc. ; 
,  t^  3  work 

2        4 
-6    +4 


MILTIPLK^ATION    AND    DIVISION. 


35 


(,hi(jiiont''^  --■  — 
2 

Q 


i.C, 


iX 

T 


8 

n 


1 

OQ   ■>      114.//'       *j8 
2          4 

8 
1 

—    • 

2.r'  +  :r^-8.r-f  1 
39 ^•^-57.1'-  7 

8     2a:^  +  ;r^-    3;*;+l 


I  )ivi(le  5.r'  - 1-  2  l)y  3.7;^   -  2x  -}-  3. 

b,if     0        0  0  0+        2--3.r^    -2.r  +  3 


1 


o 
O 


9       27 


81 


243 


3 


5+10-25-140 


0+   486 
-280 
+  225  + 1260 


2+9 


-   fc  +  1746 


1746 


n     fP      fn     +•     f      ^  I   10      25      140       1      3 

Coerts.  ot  Quotient  =  -  H •  

^  3^3'^       3'        3*       3*     3  -  2  +  3 


^     ,.     ,       5.1'^  ,   10;r^      25a;      140 
Ciuoticnt  =  — ■  + 


55^-582 


9 


27 


81       81     3:i-*'-2a:  +  3 


Bx.  13. 

Divide : 

1.  6.r'^  -h  5:^;'  -  17.7;'  -  62;^  +  IQx  -  -  2  by  2x\-\-^x-  1. 

2.  5a;«  +  6:c^-[-l  by  a;2  +  2.r+l. 

3.  a'-6«  +  5  by  «'-2«+l. 

4.  .-r  -  4a;' 7/  -  8.7;^?/'  —  17a,y  —  12//  Ijy  .i:^  -  2xy  -  3/. 

5.  (t«  -  3a^r2  +  3a^c*  —  :f«  by  «'  -  3cfc'':c  +  Sao;'  —  ^. 

6.  4.7;*  +  3.6-''  — 3.r+lby  :r''^  — 2.//-+3. 

••■■'  It  will,  in  general,  be  more  convenient  to  multiply  the  dividend  by 
siK'li  a  number  as  will  make  its  first  term  exactly  divisible  by  the  first 
term  of  the  divisor,  and  afterwards  divide  tlie  quotient  by  this  multiplier. 


30 


MULTIPLICATION    AND    DIVISION. 


7.  10.1-^  }-  5a,-*  -  90:1'''  -  Ux'  +  10a; -{- 1  by  x'  ~  9. 

8.  a^  ~  x^y  +  oc^y^  ■—  x^if  -{-  .ry*  —  if  by  ar*  —  f. 

9 .  Mill tiply  x^—  \x^a-\-^x^ a?—  4 xa^ + a*  by  x^-\-2 xa -\- a^, 

and  divide  the  product  by  a;*  —  2  ar*  a  -f-  2rf  —  a*. 

Divide  : 

10.  x^  —  ax*  -}-  hx^  —  hx"^ -^ax—1  by  a;  —  1. 

11.  6a;*+7a;*+7a;'+6a;''4-6a;  +  5by  2a,-'^  +  a;+l. 

12.  60 (a;*  +  y*) -f  91  a;y  (a:^  -  y'^)  by  12a;2- 13a;y  +  5/. 

13.  6 a;'  -  481  a;^  +  79 a;*  +  81  a;'  -  81  a;2  +  86  a;  -  481 

by  a.' -80. 

14.  6a;«-a;^  +  2a7*~2ar'  +  2a;''+19a:  +  6  by  3a;''  +  4a;+l. 

15.  a(a  +  2by-~h(2a-\-bfhy  (a~h)\ 

16.  (x-]-yy+^(x-i-yyz  +  ^(x-{-y)z'  +  z' 

hy(x  +  yy  +  2(x-\-y)z-{-z\ 

17.  10a;'*'+10a;«+10a--='-200by  a;'  +  a,-'-a;4-l. 

18.  hmx*  -{-  (hn  +  cm)  x^  -\~  cnx^  +  ahx  -\-  ac  by  bx  +  c. 

19.  Multiply  1  +  ^^ a:-  18a;'  by  1- J^a;'  +f  ar^  and  divide 

the  product  by  1 +-V-^  —  3a;^. 

Find  the  remainders  in  the  following  cases  ; 

20.  (a;^  +  3a;'^  +  4a;  +  5)-^(a;-2). 

21.  (a;*-3a;'  +  a;-3)-^-(a;-l). 

22.  (a;*  +  4a;'  +  6a;  +  8)H-(a;  +  2). 

23.  (27a;*-y*)^(3^~2y). 

24.  (3a;^  +  5a;*-3a;^  +  7a;^-5a;  +  8)^(a;^-2a;). 

25.  (5a;*  +  90 a;' +  80 a;^- 100 a; +500) -J- (a; +17). 


2 


th 
tir 


MULTIPLICATION    AND    DIVISION. 


S7 


9. 


+  2.ra-f  ^^ 
xc^  —  a*. 


^+1. 
-481 


1. 

IX  -f  c. 
and  divide 


L*). 


§  8.   The  folic 


)lcs  of 


lowing  are  exam]_)lcs  ol  an  important  use  of 
Horner's  Division  : 

1.    Arrange  or*  —  Gar*  -f-  7.r  --  5  in  powers  of  x  -  -  2. 


2 

1     -G 
o 

-{-7 
8 

-5 
o 

-  -  ^ 

2 

1     ^4 
o 

1; 

-4 

7 

2 

1    -2; 

+  2 

1;       0 

5 

Hence,  a;'-  6 a;' +  7a;-  5  --  {^x-  2)'-  5(.t;-  2)-7; 
or,  as  it  is  generally  expressed, 

''     Ga,-^  +  7i  -  5  =  y'  -  5y  -  7  if  y --  :c  -  2. 


X 


2.    Express  a;*  +  12ar'-f-47a;''  +  GGa;  +  28  in  powers  of  a,-+3. 


1 

12 

47 

G6 

28 

3 

-3 

-27 

60 

-18 

1 

9 

20 

0; 

10 

3 

3 

-18 

-  G 

1 

6 

2; 

0 

3 

3 

9 

1 

3; 

—         i 

o 
D 

-    3 

1; 

0 

Hence,  a.-*  +  12a,-^  +  47a,-^  +  GGa;  +  28  -=  y*  -  7y^-f  10, 

After  a  few  solutions  have  been  written  out  in  full,  as  in 
the  above  examples,  tlio  writing  may  be  lessened  by  omit- 
ting the  lines  opposite  the  increments  (—2  iii  Exam.  1,  and 


38 


MULTirMrATION    AND    DIVISION. 


3  in  Exam.  2),  the  multiplication  and  addition  being  per- 
formed mentally.  The  last  example,  written  in  this  way, 
would  appear  as  follows  : 

4t  66         28 

20  6       (10) 

2  (0) 
(-7) 


12 

9 

6 

3 

m 

Ex.  14. 

Express  : 

1.  x^      bx'^  -\-  3.1-      8  in  powers  of  x      1. 

2.  x^  -\- ^ x'  -\- 6  X  -]- 9  in  powers  of  a;  4- 1- 

3.  X*  -  8a;'  +  24a;'  -  32a;  +  97  in  powers  of  a;  -  2. 

4.  0;*  + 12a;^  + So;'^  — 7  in  powers  of  a;-|-2. 

5.  ?>x^  —  x^-^  4a;'  +  5a;  —  8  in  powers  cf  a;  —  2. 

6.  a;*  —  7a;^  +  11  ^'^  —  7  a;  +  10  in  powers  of  a:  —  If. 

7.  x^  —  2.t;'  —  4a;  -f-  9  in  powers  of  x  —  f. 

8.  a;'^    -  9a;'y  +  6  a;?/'  —  8?/^  in  powers  of  a;  —  2y. 

9.  x^  —  bx'^y-\-K)xy^  —  if  in  powers  oi  x  —  y. 

10.  8.2;^  -f  12a;'?/  +  10a;y'  -\-  8y^  in  powers  of  2a;  +  y. 

11.  x^  —  fa;'  +  fa;  —  t^  in  powers  of  ^x  —  ■^. 

12.  a;*  -f  8a;^  —  15a;  —  10  in  powers  of  a;  -{-  2. 


■-O-^lri, 


being  per- 
thia  way, 


CIIArTER  IT. 


Symmetry. 


2. 


1  3 


\+y- 


§  9.  An  expression  is  said  to  be  symmetrical  witli  respect 
to  two  of  its  letters  when  these  can  be  interchanged  by 
substituting  each  for  the  other  without  altering  the  expres- 
sion. 

Thus,  rt'  -\'  d?x-{-  ax^  -f-  x^  is  symmetrical  with  respect  to 
a  and  x,  for  on  substituting  x  for  a  and  a  for.r  it  becomes 
x^ -\- x^  a -\' xc^  -\- cc^ ^  which  differs  from  the  original  expres- 
sion merely  in  the  order  of  its  terms  and  of  their  factors. 
So,  also,  x^  +  i^x  -y  ah  -f-  V^x  is  symmetrical  with  respect  to 
a  and  Z>,  for  on  substituting  h  for  a  and  a  for  h  it  becomes 
x^'\-1?x-\-ha-\-a^x,  which  is  identical  with  the  given  expres- 
sion. On  interchanging  a;  and  a,  x'^-[-ti^x-\-ah-[-h'^x\)QComQii> 
d^-{-x\t-{-xh'\-h'^a\  this  is  not  the  same  as  the  given  ex- 
pression, which  is  therefore  not  symmetrical  with  respect 
to  X  and  a.  In  like  manner,  it  may  be  shown  that  this 
expression  is  not  symmetrical  with  respect  to  x  and  h. 

An  expression  is  symmetrical  with  respect  to  three  or 
more  of  its  letters  if  it  is  symmetrical  with  respect  to  each 
and  every  pair  of  these  that  can  be  selected. 

Thus,  x^-[''if-\-z^~?txyz  is  symmetrical  with  respect  to 
X,  y,  and  z,  for  it  remains  the  same  on  interchanging  x  and 
y,  or  y  and  z,  or  z  and  x ;  and  these  are  all  the  j^airs  that 
can  be  selected  from  x,  ?/,  and  z.     So,  also, 


•     \ 


40 


SYMMKTIIY. 


nnd      (x  -  n)(h     rf  \  (./:      />)(c  -  a)'  -h  (•'-      6')(a  -  />)' 
uro  oacli  syiinnctrical  willi  rospect  to  a,  />,  juid  r, 
but.       ^rV>  I  //^•-l-6•^l 

un.l      (.7;   -  (r)  (a  -  /»)'  - i-  C^:  -  ^>) (^'      ^')'  I  (•''  -  "  '0 i''  -  ^0' 
arc  not  so. 

aA  \-ac  -\'  ad  j  /;t'  {  h<i  |  cf ? 

in  Hyiniiiotrical  witli  roHpoct  to  «,  />,  c,  iiiitl  </, 

but       (ih  l  hc-\-('(l  \  da 

is  not  Rymniotriciil  willi  rospoct  to  these  l(3ttL'rs,  for  on  inter- 
changing a  and  b  it  iKH'onios 

ab~\-ao  {-cd-l'hd. 

So,  also, 

hcd-\-acd~\'ahd-\-ahc 

is  symmetrical  with  respect  to  (i,  b,  c,  and  d,  as  may  bo  seen 
at  once  by  writing  it  in  the  foi-m 


\a      0      c      a) 


An  expression  is  cyclo-symmetric  with  respect  to  three 
of  its  letters,  a,  i,  and  c,  if  it  remains  the  same  expression 
when  a  is  changed  into  5,  b  into  c,  and  c  into  a. 

Thus,  a'^i-f  Z>'^c-f  t'^«  is  cyclo-symmetric  with  respect  to 
th;)  cycle  {abc),  for  on  changing  «  into  b,  b  into  <?,  and  c  into 
a,  it  becomes  b'^c-\-c'^a-\-a}b,  which  differs  from  the  orig- 
inal expression  only  in  the  order  of  its  terms. 

(a  —  by  +  (b  —  cy  -{-(c  —  ay  is  not  symmetrical  with  re- 
spect to  a,  b,  and  c,  for  interchanging  a  and  b  changes  it 
mio(b~ay+(a~oy-\-(c-by^whk}l=-{a~by-(b-cy 
—  (c  —  ay,  and  this  differs  from  the  original  expression  in 


1 


■)(r~  ay 


for  on  inter- 


may  bo  seen 


ict  to  three 
expression 

respect  to 
and  c  into 
n  the  orig- 

al  with  re- 
changes  it 

y-(b-cy 

pression  in 


the  HignH  of  all  itn  tormH.  Hut  thin  exproHHion  in  cydo- 
HyiniiU'tiic  with  renpect  to  (nhr). 

So,  also,  {x  a)  (a  hf  \  (.7;-  />)(/>  r)'  [  (r  r)(n-  uf 
is  cyclo-HynniKftric  with  r(!spoct  to  {<ihr)  hut  in  not  coui- 
pletcly  synunelric  with  respect  to  a,  /->,  and  c. 

(nmcrally,  an  exjJresHion  is  cyclo-Hyniniotric  with  r<!Hpe(^t 

to   any   set  of  lett'.M'S,  a,  />,  c ,  A,  k,  called  the  cych'. 

{((ha hk),  if  it  remains  the  same  oxpn'ssion   when  a  is 

changed  into  h,  h  into  r,  ,  A  into  /:,  and  /•  into  <i. 

Thns,  «/>-}- ic -f- rv/ -f- c/a  and  ar  \  hd  an;  eacdi  cy(!!o-sym- 
inctric  with  respect  to  the  cycle  (abed),  l>iit  are  not  coin- 
j)let«'ly  symmetric  with  respect  (    a,  h,  e,  and  d. 

Every  expression  which  is  completely  symnn^tric  witli 
respect  to  a  set  of  letters  is  necessarily  cyclo-Hymnnitiic 
with  res])ect  to  them;. but,  as  is  seen  by  tlu;  above  exam- 
ples, an  expression  nuiy  be  cyclo-symmetric  without  being 
completely  symmetric. 

y  Principle  of  Symmetry.  An  expression  ivhieh  in  any 
one  form  is  complciely  symmetries  or  is  eyelo-symni<'.tri('^  vnOi 
respect  to  any  set  of  Icttei'S  will  in  every  other  for\ri  he  com- 
pletely symmetric,  or  he  cyclo-symmetric,  as  the  case  may  he, 
with  respect  to  these  letters. 

Thus,  ct^  -\'  h^  +  c^  —  ^ahc  is  symmetrical  with  respect  to 
a,  h,  and  c\  hence,  it  will  be  symmetrical  when  written  in 
any  other  form,  as,  for  example,  in  the  form 

^(a-\-h-\-c)[(h-cy-\-{c-ay-^{a-hyi 

Again,  {a  —  hy  -{•  (b  —  cf  +  (c  —  a)'  is  cyclo-symmetric, 
but  not  completely  symmetrical,  with  resj^ect  to  (ahc) ;  it 
will  therefore  remain  thus  cyclo-symmetric,  but  not  com- 
pletely symmetrical,  under  every  change  of  form  which  may 
be  given  it ;  for  example,  when  it  is  reduced  to 

S(a~h)(b~c)(c  —  a). 


M 


42 


SYMMETRY. 


A  symmetric  function  of  several  letters  is  frequently 
represented  by  writing  each  type-term  once,  preceded  by 

the  letter  S  ;  thus,  for  a  +  ^  +  ^+ +  ^  we  write  %a,  and 

for  ah  -\- ac -\-  ad -\- -\- he -{- hd -{- (that  is,  the  sum  of 

the  products  of  every  pair  of  the  letters  considered)  we 
write  ^ah. 


Ex.  15. 

Write  the  following  in  full : 

1.  %a'h',    ^(a-hf;    %a{h-c)]    %ah(x-c);    ^a'h^c, 

%  (a  +  h)  (c-a)  (c-h);    S  [(a  +  cf  ~  h']  ;    and 
]§«(/>  +  cy,  each  for  a,  h,  c. 

2 .  '% (the  ;  %a^h  \  ]$ a^ he ;  %(a~h)]  and  S a^ (a  —  h),  each 

with  respect  to  a,  h,  e,  d. 

Show  that  the  following  are  symmetrical : 

3.  (x  +  a)  {a  f  h)  (^  +  ^)  +  «^^,  with  respect  to  a  and  h. 

4.  {a -{- hy -\' {a  —  hy  with  respect  to  a  and  h,  and  also 

with  respect  to  a  and  —  h. 

5.  (cih  ~  xyy  ~{a'\-h  —  x  —  y)  [ah  {x-\-y)  —  xy  {a  ~\-  h)'] 

with  respect  to  a  and  ^,  and  also  with  respect  to  x 
and  y. 

6.  a\h  -  cy-^h\e~a)''-{-c\a-~hy  with  respect  to  «,  Z>,  c. 

7.  {ac  -f  Z>(i)^  -f  {he  —  ady  with  respect  to  a^  and  i^  and 

also  with  respect  to  e^  and  d^. 

8.  ,'?:''  -|-  y*  +  3  ^ry  {x^  -\-xy-{-  y'^)  with  respect  to  x  and  y. 

9.  [x'-f  +  ^xy{2x-\-y)Y-\-[y^~i^-\-^xy{2y-\-x)Y 

with  respect  to  a:  and  y, 

10.    a  (a  +  2Z>)"  -\-h{h-\-2ay  with  respect  to  a  and  h,  and 
also  with  respect  to  a  and  —  Z». 


SYMMETRY. 


4a 


frequently 
)receded  by 
'ite  Sa,  and 
the  sum  of 
3idered)  we 


");   ^a^h^c, 
—  l'^^ ;    and 

[a  —h),  each 


a  and  b. 
>,  and  also 

'^  («  +  b)] 
-espect  to  X 

ct  to  «,  5,  c. 
-nd  b^,  and 

a;  and  y. 
nd  i,  and 


11.  ah  \[(a  -i-r)(b-{-  ^)+  2^(a  +  i)]'^  -  (a-  rf^b  -  ^y| 

with  respect  to  a,  b,  c. 

12.  (eb""  +  iV'  +  6'^a'  -f  2«/>6'(«  +  Z»  -I-  r)  with  respect  to  cr^>, 

be,  ca. 

With  respect  to  what  letters  are    the    foUowiiiii;   sym- 
metrical ? 

13.  xyz  +  5 ary  -f  2 {x^  -|-  y'). 

14.  2 (a' a;'  +  6'y')  -  2ab  {xij  ^  />?/  j  ru). 

15.  (/^-A7  +  4/(/+/0'-K2/A--2y7. 

16.  (:t^  +  ?/)(^-2;)(y-2;)-a;7/2;. 

17.  a^b^  +  i'c^  +  .'a'  -  2abc{a  +  Z»  -  c). 

18.  x^ -  y«  +  2"  -  3 (o.-^  -  /)  (,if  -  r)  {^  - f-  X-). 

19.  (a  +  Z>)M- («  +  ^)' +  (^^ -- ^)'. 

20.  (a  +  i)*  -I-  (a  -  c)'  -f  (/^  -f  c)*  - h  (^/^  -  h  <^'f- 

21.  (a  +  Z*)*  -\-{a-  cy  +  (5  +  6')'  I-  (a  -|-  6-)*  +  (^"  -  b)*. 

Select  the  type-terms  in  : 

22.  a'  +  2a^'-f  Z*'4-2ic4-c'  +  2ca. 

23.  a (b'-  c')  +  b  (c'-  a')  -f  c (a' ^  b')  +  (a  -(-  b)  (b-\- c)  (c-{-a). 

24.  a{b-{-cy-]-b(c-\-ay-]-c(a-\-by-l2abc. 

Write  down  the  tyjji^-terms  in  : 

25.  {x-{-yy]  {x-~yy\  {x -\-yf  -  x^ —  7/. 

26.  {x-{-yy^{x--ijy]  {x  +  yy-{x-~yy. 

27.  {x-\-y-\-zy]  (x-y-zy. 

2S.    (a +  bi- a -j-dy-,  (a' i-b'  + a'  \-cPy. 

29.    (ct  +  Z*y -f  (^  +  c)' +  0' +  «)'. 


44  SYMMETRY. 

§  10.  In  reducing  an  algebraic  expression  from  one  form 
to  another,  advantage  may  be  taken  of  the  principle  of  sym- 
metry ;  for,  it  will  be  necessary  to  calculate  only  the  type- 
termSf  and  the  others  may  be  written  down  from  these. 

Examples. 

1.  Find  the  expansion  oi{a-\-h-\-c-[-d-\-e-\- y. 

This  expression  is  symmetrical  with  respect  to  a,  h,  c, 

;  hence  the  expansion  also  must  be  symmetrical, 

and,  as  it  is  a  product  of  two  factors,  it  can  contain 

only  the  squares  a^  5^  c^,  ,  and  the  products  in 

pairs,  ah,  ac,  ad,  ,  be,  bd,  ;   so  that  a^  and  ah 

are  type-terms. 

Now  (a -{-hf  =^a^  '\-2ah-\-h'^ ;  and  the  addition  of  terms 
involving  c,  d,  e, ......  will  not  alter  the  terms  a^-\-2ah, 

but  will  merely  give  additional  terms  of  the  same 
type.     Hence,  from  symmetry  we  obtain 

(a-\-h-{-c-}-di-e+ y=  a''-{-2ab-\-2aci-2ad+2ae-{- 

i-h''     i-2he+2hd-}-2be  + 

+c'      \-2cd'\-2ce-}- 

+  d^    -\-2de-\ 

.  +6'     + 

This  may  be  compactly  written 

i^aJ  =  %a^-\-2^ah. 

2.  Expand  {a  +  h)^. 

(i.)  The  expression  is  of  three  dimensions,  and  is  sym- 
metrical with  respect  to  a  and  h.  .  _  , 
(ii.)  The  type-terms  are  a^,  a^b. 

Hence,  (a  -f  by  =  a^  -{-  h^  -i-  71(0^  h  -{-  b'^ a),  where   n  is 
numerical. 

To  find  the  value  of  n,  put  a  =  b  =  l,  and  we  have 
(l  +  l)^--l  +  l  +  w(l  +  l).     .-.71  =  3. 


SYMMETRY. 


m 


)m  one  form 
iple  of  sym- 
ly  tlie  t^2^e- 
Q  these. 


.....y. 

3t  to  a,  h,  c, 
ymmetrical, 
can  contain 
products  in 
b  a?  and  ah 

ion  of  terms 
■ms  a?-\-2ah, 
of  the  same 

f2ae-f 

^26e  + 

\-2ce-\- 

2de-\ 

e'     + 


md  is  sym- 

^here   n  is 
have 


3.  Expand  {x-\-y-\-  z)'. 

This  is  of  three  dimensions,  and  is  symmetrical  with 
respect  to  x,  y,  z.     We  have 

(a:  +  y  +  z>^  =  [(^  +  3/)  +  z]-'-(:r4-2/)^+ 

which  are  type-terms,  the  only  other  possible  type- 
term  being  xyz. 

Now,  since  the  expression  contains  2>x^y,  it  must  also 
contain  2>x^z\   that  is,  it  must  contain   2>x^{y-\-z). 

Hence, 

{x-\-y-{zf=    x'  +  ^x'iyi-z) 
■\-f  +  ^y'{z^-x) 
+  2'+3z^(:r+2/) 
-{■n{xyz), 

where  n  is  numerical,  and  may  be  found  by  putting 
x  =  y  =  z  =  l  in  the  last  result,  giving 

(l  +  l  +  l)^=l  +  H-l-f3(l  +  l)  +  3(l-fl) 
+  3(1  +  1)4-^. 
Hence,  w  =  6. 

4.  Similarly,  we  may  show  that 

{a-\-h-\-c-\~df=^     (c'-\-'^a^{h-\'C-\-d)-\-Q>hcd 

+  Z»-^  +  35^  (c  +  cZ+ rt)  +  6c(Za 
+  c^  +  3c^  ld-\-a-\-h)  +  6dah 
+  d'+^d^la-\-h-{~c)-i-(jahc. 

5.  Expand  (a  +  ^^  +  ^  + )'• 

The  type-terms  are  a',  cfh,  ahc. 

Expanding  (a-{-h  -{-cf^  we  get  a^-{-2>a}b-\-(jahc-\- 

Hence,  by  symmetry,  we  have 
{%df-^%a^-\-?>^aV)-\- (Stale. 


\ 


46 


SYMMETRY. 


6.  Simplify  (a  i- Ij  -  2cy -}- (b +  c  -  2ay  +  (c  -f-  a  -  2^)1 

This  expression  is  symmetrical,  involving  terms  of  the 
types  a^  and  ab.  Now,  a^  occurs  with  1  as  a  coeffi- 
cient in  the  first  square,  with  4  as  a  coefficient  in  the 
second  square,  and  with  1  as  a  coefficient  in  the  third 
square,  and  hence  6  a^  is  one  type-term  of  the  result ; 
ab  occurs  with  2  as  a  coefficient  in  the  first  square, 
with  —4  as  a  coefficient  in  the  second  square,  and 
with  —4  as  a  coefficient  in  the  third  square,  and  hence 
—  6ab  is  th  second  type-term  in  the  result.  Hence, 
the  total  result  is  6  (ci^  -\-  b"^  -\-  c^  —  ab  —  bc~  ca). 

7.  Simplify  (cc  ~\- 7/ -\- zf  -{- (x  —  y  —  zf -\- (y  —  z  —  xf 

-{-(z  —  x  —  y)\ 

This  is  symmetrical  with  respect  to  x,  ?/,  z ;  and  the 
type-terms  are  o:^,  2>x'^y,  (Sxyz: 

(i.)  x^  occurs  in  each  of  the  first  two  cubes,  and  —a^m 

each  of  the  second  two  cubes ;  therefore,  there  are  no 

terms  of  the  type  x?  in  the  resulL. 
(ii.)  ^x^y  occurs  in  the^rs^  and  third  cubes,  and  —^xSj 

in  the  second  and  fourth ;  therefore,  there  are  no  terms 

of  this  type  in  the  result, 
(iii.)  Q>xyz  occurs  in  each  of  the  four  cubes;  therefore, 

2^xyz  is  the  total  result. 

8.  Prove (a^-\-b''-\-c^+d'')  {w''-^x^-\-y''-]-z'y-{aw^bx-{-cy-\-dzy 

=  (ax — bwf  -f  {ay  —  ciuf  +  {az — dwf  -\-  (by — cxy 
+  lbz-dxy  +  (cz--dyy. 

The  left-hand  member  (considered  as  given)  is  symmet- 
rical with  respect  to  the  pairs  of  letters,  a  and  w,  b 
and  X,  c  and  y,  d  and  z ;  that  is,  any  two  pairs  may 
be  interchanged  without  affecting  the  expression.  As 
the  expression  is  only  of  the  second  degree  in  these 


SYMMETRY. 


47 


terras  of  the 
L  as  a  coeffi- 
Sicient  in  the 
k  in  the  third 
)f  the  result ; 
?  first  square, 
square,  and 
re,  and  hence 
5ult.  Hence, 
'~ca). 

-  z  —  xf 

',  z ;   and  the 

?,  and  —cc^  in 
there  are  no 

and  —Sx^y 
are  no  terms 

s;  therefore. 


-hx-{-c7/-\-dzy 
hy—cxy 

)  is  symmet- 
,  a  and  w,  h 
)  pairs  may 
)ression.  As 
ree  in  these 


pairs,  no  term  can  involve  three  pairs  as  factors ; 
hence,  the  type-terms  may  be  obtained  by  consider- 
ing all  the  terms  involving  a,  h,  w,  x\  these  are  aW, 
a'^^•^  h'^iv\  b^xK  —c^vj^,  —U^x^,  —2ahwx,  and  are  the 
terms  of  (ax  —  bwy,  which  is  consequently  a  type- 
term.  From  {px~hwy  we  derive  the  fivo  other  terms 
of  the  second  member  by  merely  changing  the  letters. 

9.    Prove  that  {x^  —  yzf  +  {^if  —  zxf  -f-  {z^  —  xyf 

—  3(a;'^  —  yz){y^  —  zx) (f  —  xy)  is  a  complete  square. 

The   expression   will  remain   symmetrical   if  {x^  —  yz) 
(y"^  -zx){z^  —  xy),    instead    of  being   multiplied   by 

—  3,  be  subtracted  from  each  of,  the  preceding  terms, 
thus  giving 

{x^  —  yz)  [{i?  ~  yzf  —  iy^  —  zx)  {z^  —  a:?/)] 

+  (y'  -  2^')  W  ~  z^y  -  (2'  -  ^y)  {^'  ~  y^)] 

H-  (2'  —  xy)  r(z'  —  xyY  -  (.r"  —  yz)  {f  -  -  zx)  ] 

—  {x}  —  yz)x{s?  +  2/^-^2^  —  Zxyz)  + 

=  (^'^  +  2/^  +  2'  —  Zxyz)  (x'  -j-  2/'  -f  2;''  ~  3  xyz). 


Ex.  16. 


Simplify  the  following : 


1. 
2. 
3. 

4. 
5. 

6. 

7. 


a  +  b-{-cy-{-(a  +  b- cf  -{-(b^-c- ay  +  (c-\-a  - 

a-b-cy  +  (b  —  a-cy-^(c-a-  by. 

a  +  b  +  c  —  dy  +  ib  +  ci-  d  -  ay -\- {c  -{-d+a- 
-i-(d  +  a-{-b-cy 

(c  -\-b  -\-  cy  —  a(b-\~c  —  a)~-b(a-\-c  —  b)  —  c(a-{-b 

^'-{-y  +  z  +  ny-\-(x  —  y~z  +  ny  -f  {x  -  ?/  +  2;  - 
+  (a;  -f-  y  -  2;  —  ny. 

r«  -f  Z>  4-  6f  +  (a  +  i  -  cf  -f  (5  +  c  -  ay-\-{c-\-a~ 

x-2y~  ^zy  -f  (2/  -  2s  -  2>xy  -f-  (2  -  2x  -  3y)^ 


by. 
-by 

ny 
by. 


48 


SYMMETRY. 


8.  (ma  -[•  nh  -\-  rcy  —  (ma  -f-  nb  —  rc)^  —  (nh  +  re  —  ma)^ 

—  {re  +  wa  —  nhf. 

9.  a  (h  +  c)  (}?  -f  c'  -  a')  -\-h(c-\-  a)  (c'  +  a'  -  b') 

-{-c(a  +  b)(a'  +  b'~c'). 

10.  (ab-\-bc-i-ca)'^ —  2abc(a-{-b-{-c). 

Prove  the  following : 

1 1 .  (ax  -{-  by  -{-  czf  +  (bx  -f-  ci/  +  cizy  +  (ex  +  a?/  +  bzf 

-f-  (ax  +  c?/  +  Z>z)^  -f  (d?a:  -f  ^y  +  «2)'^  -f-  (ia:  4"  «y  +  czy 
=  2  (f/  +  b'  +  c^)  (a,-  +  f  +  2'0 
+  4  (ab  -{-be  -\-  ea)  (xy  -\-yz-\-  zx). 

12.  (a-i-b-^ey  +  (b  +  e-ay+(ei-a-  &)*  +  (a  +  6-c)* 

=  4  (a*  +  6*  -f  c*)  +  24  (a^fe^  -}-  ^'^c'^  +  c'a'). 

13.  (a  +  ^>  +  c)*  =  Sa*  +  4Sa='i  +  6:Sa^52+12^a^^>c. 

14.  ('S,ay  =  ^a*  +  ^^a'b  +  (j:^d'b''  +  12'S,a''be-}-24:^abccl 

15.  (^2,^^  -f  c'^)3  4-2(aZ»  +  5c  +  m)^ 

—  3  {a"  +  b'  +  c')  (ab  -{-be-}-  eay 
=^(a'  +  b'  +  c^-Sabey. 

16.  (a-  by  (b  -  ey  +  (b-  ey  (e  -  ay  +  (e  -  ay  (a  -  by 

=  (a"  +  b^  +  c''-ab  -  ac  -  bey. 

17.  (2a-b-cy(2b-c-ay-\-(2b-e~ay(2c-a-by 

+  (2e-a-by(2a-b-cy 

=  9(a''  +  b^-{-c^-ab-be-eay. 

18.  (ar'-i-2brs  +  es^)(ax''  +  2bxy-{-ey'') 

—  [arx  +  b  (ry  -\-  sx)  -f-  esy]  '^  =  (ac — Z>^)  (ry  —  sa?)''. 

19.  (a''  +  ab  +  b'')(e''  +  ed-i-d^) 

=  (ac-{-ad-\-bdy-\-(ae~\-ad^bd)(be—ad)-{-(be—ady. 

20.  Show  that  there  are  two  ways  in  which  the  given 

product  in  the  last  example  can  be  expressed  in  the 
form  p^  ^pq-{-  q^,  and  two  ways  in  which  it  can  be 
expressed  in  the  form  ^:)'' —jj^' -f  2'^ 


THEORY    OF    DIVISORS. 


m 


)-{-rc  —  may 


-  mj  +  hzY 
'}X-\-ay-\-czf 


^{a-\-h-cy 

l'!Za^  be. 

^-]- 2^%  abed 


ay  {a -by 

{2e-a-by 


ry  —  sxy. 

-\-(be-ady. 

the  given 
issed  in  the 
h  it  can  be 


21 .  6  (i^;'  +  ^^'  +  f  +  z'y  =  (w  -\-  xy  +  (w-  xy  -[-  (^o  h  y)* 

+  (^  -  y)*  +  (^  -f  zy  +  (10  -  zy  +  (x  +  y)^-H  (a;-y)* 

+  (^  +  2/  +  (^-2)*  +  (y  +  z)H(y-2)\ 

22.  ^[(a-{-b-\-ey-i-(a-b--cy  +  (b~c~ayi-(e--a~by] 

=  i[((^+b-}-ey+(a-b-cy-{-(b-e-ay-{-(c-a-by] 
X  i  [(a + ^'  4- ^)'+  (a  -  6  -  cy+  (b -e-ayi-  (c-a-by]. 

Theory  of  Divisors. 

Any  expression  which  can  be  reduced  to  the  form 

ax''  +  bx*"-^  +  cx""-^  + +  hx+k, 

in  which  w  is  a  positive  integer,  and  a,  b,  e, ,  h,  h,  are 

independent  of  x,  is  called  a  Polynome  in  x  of  degree  n. 

The  expressions  f(xy,  F{xy,  <f>  (a;)*",  are  used  as  general 
symbols  for  polynomes;  the  exponents  n  and  m  indicate 
the  degree  of  the  polynome. 

Theorem  I.  If  the  polynome  f{xy  be  divided  by 
X  ~  a,  the  remainder  will  be /(a)**. 

Cor.  1.  fixY—fiay  is  always  exactly  divisible  by  x—a. 

Cor.  2.  If  f(ay  =  0,  f{xY  is  exactly  divisible  hy  x  —  a\ 
that  is, /(a;)"  is  an  algebraic  multiple  oi  x  —  a. 

Cor.  3.  If  the  polynome  /(^)",  on  division  by  the  poly- 
nome <ji  (xy,  leave  a  remainder  independent  of  x,  such 
remainder  will  be  the  value  oi/(xy  when  <^  (.r)"*  =  0. 


Examples.     Theorem  I. 


1. 


12 


Find  the  remainder  when  a^ —7x* -\-lBx^ —IQx"^  -{-dx 

is  divided  by  a;  —  5. 
The  remainder  will  be  the  value  of  the  given  polynome 

when  5  is  substituted  for  x.     (See  §  3.) 


k  "V 


60 


THEORY    OF    DIVISORS. 


1 

-7    +13 
5        10 

16+9 
15        5 

-12 

20 

1 

-2          3 

-1        4; 

8 

Hence,  the  remainder 
remain  d 


is  8. 


2.    Find  the 


when  (x  ~  a)'  +  (x  —  bf  +  (a  +  hf 


is  divided  by  a;  +  «. 
For  X  substitute  —a,  then 

(-  2ay  +  (~a-hf+  (ai-by  =  ~  8 


a' 


3.  Find  the  remainder  when 

ar' +  a' +  5' +  (:r  +  a)  (a;  +  Z*)  (rt  +  Z>) 
is  divided  by  x-}-  a-\-b. 

For  X  substitute  —  (a  +  h),  and  we  obtain 

-  (a  +  ^•y  +  tt'  +  b^  +  ^ib  (a+  5)  -----  2a6(a  +  b). 

See  Formula  [6]. 

4.  Find  the  remainder  when 

(-'  +  2ax -  2a'^)^  (rc^  -  2ax -  2a'^)  +  32  (x  -  a)*  (.r  +  a)* 
is  divided  by  x^  —  2a\ 

x"^  —-2a^  may  be  struck  out  wherever  it  appears. 
This  reduces  the  dividend  to 

(2  axf  (-  2  ax)  +  32  (x  -  a)*  (x  +  -.)* 

=  -16a*;r*  +  32(a;'^~a^)*. 

In  this  substitute  2a^  for  x^,  ar.d  it  becomes 
-64a«  +  32a«  =  -32a^ 
which  is  the  required  remainder. 

Ex.  17. 

1.  Find  the  remainder  when  3.'c*+  60x^-{-  54a;^—  60:i:  +  58 

is  divided  by  :r  +  19. 

2 .  Find  the  remainder  when  joa;^  —  Sqx"^ + 3  7'a: — s  is  divided 

by  a;  —  a. 


li 
1 

i: 
i; 


J 


12 
20 

8 


y+(a-^hy 


b{a^b). 


a)*  {x  -f  of 


ears. 


THKORY    OF    DIVISORS. 


61 


3.  What  number  added  to 

4ar^+  34a;*  +  58r^  +  21a;^  -  123a;  ^-  41 

will  give  a  sum  exactly  divisible  by  2a; -f- 13. 

4.  What  number  taken  from 

lOa;'"'  -  200;" -  10 a;"  -  0.89a;*  ~  8.9:r'^  +  20 

will  leave  a  remainder  exactly  divisible  by  10. i"    11  ? 

Find  the  remainders  from  the  following  divisions  : 

5.  (.^'+l)^-a.•^-^-a;+l;  {x'^[-a-\-^f~{x-\-a-\-Vf^x\-2. 

6.  x'''\-if-^x--y\  .^•■'''-f  ,"*-^a;+y;  a;'^"+^+?/'''+^-H-a.-4-7/. 
7  -    (a;  -f  1)^  +  x^  -f-  {x  -Vf-^x  -  2. 

8 .  (.r  -  of  {x  +  (0'  +  C^-'  -  2  by  -=-  x'  -  h  ^'^ 

9.  (^x"^ -{- ax -\- o?)  {x"^  —  ax -\- a?) 

-  (.1-2  -  3  aa;  +  2  a')  (a;'^  +  3  a.2;  +  2  a')  -:-  x""  +  2  rtl 

10.  (9a^  +  66«^>  +  4^*')  (9a^  -  6a/^  +  4^''0  (81a*  -  366*^5^  - 1^  IGZ**) 

-^  (3a -2^)1 

11.  a^  (a;  —  a)'"'  -f  ^^  (-^   -  ^^)"'  -^  -'^  ~-  a  —  i. 

1 2 .  («a;  -f  /;?/)■'  +  a^  v/  +  ^•''  ar*  —  3  ahxy  (ax  +  Z»y) 

-^(a  +  5)(a;  +  y). 


13.    a,-''+a^+  Z*'—  3«Z'a; 
also,  -f-  a;  —  ct  —  h. 


X 


a -}-  S  ;    also,  ^ x-{-  a~h] 


14.    Any  polynome  divided  by  x—1  gives  for  remainder 
the  sum  of  the  coefficients  of  the  terms. 


-60  a; +  58 
is  divided 


ExAMrLES.     Cor.  1. 

1.    a;^ -}- 7/*^  is  exactly  divisible  by  a; -f- y. 

In  ''a^~a^  is  exactly  divisible  by  x—a,''  substitute  -~y 
for  a. 


52 


THEORY    OF    DIVISORS. 


2.  ma^—px'^-\-q.r-{'Vi{-2^'\'^  ^^  exactly  divisible  \)y  a:+l. 
This  may  be  written 

(wr»  -px'  +  qx)  -  [m  (^ly  -;;(  ^  1)'-|-  Q(~  1)] 
is  exactly  divisible  by  x  --  (—  1). 

3.  (x^  +  6.r?/ 4-  4:1/ f  ■  h  (x'  -i-  2x1/  +  ^V^f  is  exactly  divis- 

ible by  (a;  f  2  y^. 

For  {x"^  +  6a;?/  +  Aiff  —  (—  x"^  ~2xy-~  Aiff  is  exactly 
divisible  by  {x^  Ar^xy -}- 4'if)  —  {—  x^  —  ^xy  —  ^ y^), 
which  is  2  (V  +  4  a;y  +  4 y)  =-  2  (a;  +  2y)l 


3. 

1       . 

i 

\               1 

i              ■' 

4. 
5. 

6. 

i 

7. 

1 

8. 

iji 

9. 

i 

10. 

11. 

.'" 

12. 

13. 

Ex.  18. 

Prove  that  the  following  are  cases  of  exact  division  : 

1 .  a;''"+'  H  -  ?/'"+^  -^x-\-y\  x"""  -  ?/''*  -^  x  +  y. 

2.  a;'''  +  y"'-i-a;*  +  y*;  r"*  + /" -f- a;'' +  / ;  also, -^  ^t'^"  +  y" ; 

also,  -^  a;^  +  y^ 

(«:r  +  hyf  +  (^o;  +  ayj  -f-  (rt  +  Z»)  (:c  +  ?/). 

{ax  -f  %  +  cz)*''  —  (i:^;  -\-  cy  -\-  azY 
-^(a~h)x-^  (h  —  c)  y  -\-  (c  —  a)  z. 

(2?/  -  xy  ~(2x  —  ?/)"  -^  3 (y  —  a;). 

(2y  -  0;)'^"+^  +  (2  a;  -  y)'''+^  -^  y  +  ^•. 

(7)iy  —  ?i.r)^  —  (??i:r  —  7iyy  -f-  (7?i  +  7i)  (y  —  x). 

(x  +  yy  +  (x~yy-^2(x'  +  f). 

(x'  -\-xy  +  yj  +  (x'  -  xy  +  yj  -  2 (x-  +  /). 

(a  +  Z^)«  -  (a  -  ^')' -^  2  Z»  (3 «^  +  5'0. 

(x'  -{-5hx  +  by  +  (:c^  -bxi-  bj  -^  2  (x  +  by. 

(^  -!_  5)4«+2  ^  (ft  _  5)4n+2  -^  2  (a^  +  Z^'^) . 

[xr^  +  3.^'?/ (a; -y)  -  y'^J'H-  [a;'  -  9a;y (a;  -  y)  -  y'Y 
^2(x-yy. 


TJIKUUY    or    DiVlSUllS. 


68 


14.    3r'      bx'  \-^x-2-i'X—l. 

16.  Any  jiolynome  in  ju  is  divisible  by  x—l  when  the 
sum  of  the  coefficients  of  the  terms  is  zero. 

16.  Any  polynomo  in  x  is  (livisil)le  by  x-]-l  when  the 
sum  of  the  coofhcionts  of  the  even  powers  of  x  is 
equal  to  the  sum  of  the  coeiiicients  of  tlie  odd  pow- 
ers. (The  constant  term  is  included  among  the 
coefficients  of  the  even  powers.) 


1  vision  : 


Examples.     Cur.  2. 

1.  Show  that  a(a-{-2by  —  b(2a-^hy  is  exactly  divisible 

by  a-\~b. 

By  Cor.  2,  the  substitution  of  —  i  for  a  must  cause  the 
polynome  to  vanish. 

Substituting,  a{a—2ay-\-a(2a  —  ay=—a*-\-a*=0. 

2.  Show  that 

{ab  —  xijf  --{a-\-b  —  x  -  y)  [ab  {x  -\-y)  ~ xy  (a  +  b)] 
is  exactly  divisible  by  (x  —  a)(y  ~  a),  also,  by 
(x-J)(2/-^i). 

For  X  substitute  a,  and  the  expression  becomes 

(ab  -  ayf  —  (b-y)  [ab  (a  -\- y)  ~  ay  {a -\-  b)] 
=  a\b-yy-{b~y)[a\b-y)]  =  0. 

The  expression  is,  therefore,  exactly  divisible  by  x—a. 
But  it  is  symmetrical  with  respect  to  x  and  y,  hence 
it  is  divisible  by  y  —  a;  and,  as  x^a  and  y—a  are 
independent  factors,  the  expression  is  exactly  divisi- 
ble by  (x  —  a)(y  —  a).  Again,  the  given  expression 
is  symmetrical  with  respect  to  a  and  b ;  hence,  mak- 
ing the  interchange  of  a  and  b,  the  expression  is 
seen  to  be  divisible  by  (a;— ^)(?/  — 5). 


M 


TIIKOKY    Ub'    DIVISORH. 


3.  Show  that  G  {(V'  -j-  h'  -f-  (/>)     5  (a»  +  //  -^  cO  (a^  +  //'  -f  c'j 

is  exactly  divisiWo  by  a-^-b-l-c. 

For  a  Hubstitute  —(b-\-c),  und  tlio  result,  which  would 
be  the  remainder  were  the  division  actually  per- 
formed, must  vaniHli. 

-b[~(b  +  cy  +  //    I-  .-^J  [(/>    1-  r')^    h  b'  +  r.^] 
--  6  [-  (b  +  c)"^  +  ^''  +  c']  H-  30  <^c-  (bi-c)  (//  f  Ac  f  c'). 
See  [1]  and  [G]. 

The  expauHion  being  of  the  fifth  degree,  and  symmetri- 
cal in  b  and  c,  it  will  be  sufficient  to  show  that  the 
coefficients  of  b^,  b*c,  Pc^  vanish,  the  coefficients  of 
Pc^,  be*,  &  being  the  coefficients  of  the  former  terms 
in  reverse  order.  Calculating  the  coeffictients  of  these 
type-terms,  we  get 

which  evidently  vanishes.     Hence,  the  truth  of  the 
proposition. 

4.  Ifa+5-f«?--0, 

i(a'^+^>>'+c')--l(«M-^>'  +  ^')Xi(^'  +  ''^'  +  ^''). 
In  the  last  example  it  has  been  proved  that  the  differ- 
ence of  the  quantities,  here  declared  to  be  equal,  is  a 
multiple  of  a+^  +  ^>  that  is,  in  this  case,  a  multiple 
of  zero.  Hence,  under  the  given  condition  they  are 
equal. 

Ex.  19. 

Prove  that  the  following  are  cases  of  exact  division  : 

1 .  {ax  -  byj  +  {bx  -  ayf  -  {a?  +  b^)  {p?  —  7/)-^a,b,  x,  y , 

a-\-b,  x—y. 

2.  a3^~{p^-\-b)x''^b'' 


ax—  h.     (Substitute  ax  for  b.) 


TIIKOUY    OF    DlVl.SOliS. 


55 


d'  +  //'  -f  c^) 

/liicli  would 
jtiially  per- 


'•1 

I  symmotri- 
low  that  the 
efficients  of 
jrmer  terms 
eiits  of  these 

■'+ ). 

ruth  of  the 


the  differ - 

equal,  is  a 

a  multiple 

)n  they  are 


rision  : 
a,  h,  X,  y, 

\ax  for  h.) 


g  r  {n:r-\-hi/f  -  (a     b)  (./;  -f  z)  {ax^lnj)  -f  ^a  -hfxz  -^-:f  +  y. 

4.    'iSa'^ji?  -4inx^  -10(1X1/  — S(t^xi/\- 2x^1/ -{-ljy^~i- 2 cix—y. 

6.    1.2n*a;-  5.4D4a\x'''  +  4.8aV  +  0.Ut«.6-'  -  x" 
-iOfjax~2x\ 

6.  .-f«  +  .r V  +  .f V  1/  ^A-^  +  y. 

9.    a(rt  +  2/>)'-Z>(/^  +  2ay-f-a-^>,  also-^a  |  />. 

11.  a(b~  cf  +  i (<,•  -  a)'  -h  6' (a -  by  -i-  (a  -  b),  (b  -  f), 

(c  -  a). 

12.  «X6  -  c)  -}-  ^='((7-  a)4-^'(«-  ^)  -^  (^*  -  ^).  C^^'  -  ^0» 

(c?  —  a). 

13.  a'  ib-c)^  b'  (c  —  n)  +  c*(a  --b)-^(a-  b),  (b  -  c), 

(c  -  a). 

14.  (a  -  /^)'  (c  -  fO'  +  (*  -  '^)'  (d  -  af  ---  (c?  -  by  (a  -  -  cy 

-J-  (a  —  ^),  (6  —  c),  (c  —  d),  (d  -  ■  a). 

+  (c-  ayb']  -  [(«-  byc  +  {b  -  cya  +  (c-  a/Z*]^ 
-i-(a  —  b),  (b  —  c),  (c  —  a). 

16.  (a:  +  y)(2/  +  z)(z  +  a:)  +  a;?/z-^a;+?/  +  z. 

17.  abid"  -  h')  +  bc{b''  -  c^)  +  ^(c^  -ce)-^a^-b  -f-  c. 


18.    (a^>  -be-  cay  -  o^h"  -  b'  c' 


&c(^  -^a 


-Vb 


19.  {a  +  2by-\-  (2b  ~'dcy-(^c-  ay-i-a'  +  Sb'-27c' 

-^  a -{-2b  — 3c. 

20.  a^b^  4-  6'c''  +  c^a'  -Sd'b'c' ^ab -{-bc  +  ca. 


^ 


I 


56 


THEORY    OF    DIVISORS. 


Examples.     Cars.  3  and  2. 

1.    Find   the   value   of  4:X^ -\- Six"  -  bx"  ^  2^x^^  when 
2a;'' -=3:?;- 4. 

Since  2^'*  —  3^"-|-4  =  0,  we  have  simply  to  find  the 
remainder  on  division  by  2:r'*  — 3a;  +  4;  and,  if  it  is 
independent  of  x,  it  is  the  value  sought,  Cor,  3. 


4    0 

9 

5 

23 

6 

3 

6 

9 

15 

-  3 

4 

-8 

-12 

20 

4 

2 

2     3 

5 

-  1; 

0 

10 

Hence,  the  required  value  is  10. 

2.    What  value  of  c  will  make  x^  —  bx^-^-lx  —  c  exactly 
divisible  by  :r— 2. 
If  2  be  substituted  for  x,  the  remainder  must  vanish, 
Cor.  2. 

1     -5        7    -c 
2-6        2 


2 


1     -3         l;2-c? 
Hence,  2  —  c  =  0,  or  c  =  2. 

3.    What  value  of  c  will  make  ^x^  —  bx!^-\-coi^~20x'^-{-l^x 
—  5  vanish  when  2x'^  =^  3;r  —  1. 

By  Cor.  3  the  remainder  must  vanish  when  the  given 
polynome  is  divided  by  2x^  —  ^x  -\-\.  We  may 
divide  at  once  and  find,  if  possible,  a  value  of  c  that 
will  make  both  terms  of  the  remainder  vanish ;  or, 
we  may  first  express  ca?  in  lower  terms  in  x,  and 
then  divide  and  find  the  required  value  of  c  from 
the  remainder. 


THEORY    OF    DIVISORS. 


57 


x-\-Q  when 

to  find  the 
and,  if  it  is 
Cor.  3. 


-  c  exactly 
nust  vanish, 


20a;'4-19;r 

the  given 

"We  may 

le  of  c  that 

ranish ;  or, 

in  X,  and 

of  c  from 


3 
-_  o 

*'st  Method  (see  page  31). 
6    10    Ac            160 
18   24  126' +  36 
- 12     -  16 

304 
36^-420 
-  8c     24 

-160 

24  c +  280 

6/  8,  4^  +  12,  12^-140; 

28  d?- 140 

-24^+120 

Hence,  28^=140  and  24:  c-^  120.    Both  of  these  are  sat- 


isfied by  c  =  5. 


Second  Method. 


x^  - 


^\x(Zx-l)^^x'~\x 


3 


X 


l^x 


3 
¥• 


=  %{^x-l)-\x  =  2\x 

Substituting  for  coi?  in  the  given  polynome,  it  becomes 

6a:^-  5a;*  -  20:^=^  +  (If  ^  +  19).r  -  %c  -  5. 

Divide  and  apply  Cor.  3. 

6-10    0  -160  28c +  304  -24c -160 
3      18   24    36     -420 
2         -12  -  16     -  24        280 


% 


12 


140;  28c -140  -24c +  120 


We  thus  obtain  the  same  remainder  as  by  the  former 
method,  and  consequently  the  same  result.  A  comparison 
of  the  two  methods  shows  that  they  are  but  slightly  differ- 
ent in  form,  but  the  second  method  shows  rather  more 
clearly  that  c  need  not  be  introduced  into  the  dividend  at 
all,  but  the  proper  multiples  of  it  found  by  the  preliminary 
reduction  can  be  added  to  or  taken  from  the  numerical 
remainder,  and  the  "true  remainder"  be  thus  found,  and 
c  determined  from  it. 

Ex.  20. 

Find  tlie  value  of : 

1.  X^ —  '6x^ -\-4x'^—2>X-\-4i,  ^lYQli  x^^x—\. 

2.  x""  -2x'  ~  4x^  -^l^x"  ~\\x      10,  given  {x-\y^2. 


58 


THEORY   OP   DIVISORS. 


3.  2x'-1  x'+12ar'~llx'^{^2x~5,  given  (x-iy-\-  2^0. 

4.  Sx^-{-nx^-{-10x'+7x''+2x+S, 

given  .r'+S^^— 2:^  + 5  =  0. 

5.-(jx'''i-9x^-lGx'-5ar'~12x^-ex-]-(jO, 
given  3  a;* -f  a;  —  4  =  0. 

What  values  of  c  will   make   the   following  polynome 
vanish  under  the  given  conditions : 

6.  x'  +  ISx^  i-  2(jx''  +  b2x  i-  8c,  given  .^  +  11  =  0. 

7.  x'~2x^  —  9x''  +  2cx--U,giYei\^x-\~7^0. 

8.  X*  —  Ax^  —  a;^  -f  16a;  +  6(?,  given  x^  =  x-{-G. 

9.  2x'  —  10^2  +  4:CX  +  G,  given  x'  +  3 -=  3rr. 

10.  2x* -h  a;'  -  7  cx^  -\-Ux  +  10,  given  2x  =-  5. 

11.  4:x'  +  ex""  +  110a;  -  105,  given  2x''  -  5a;  +  15  .  -  0. 

12.  3.T'^-lGa;*  +  ca;'-5a;''-114a;-h200,  given  a;'  =  3a;-4. 

13.  What  values  of  jO  and  q  will  make 

x*-\-  2x^—10x'^~2^x~\-  q  vanish,  if  a;^  —  3  (x  —  1)  ? 

14.  What  values  of  p  and  q  will  make 

a''  -  5  a^»  +  10  a«  -15a^  -f-  29  a*  ^   pa'  +  q  vanish,  if 
(a'~2y  =  a'-^? 


Theorem  II.    If  the  polynome  /(a;)"  vanish  on  substi- 
tuting for  X  each  of  the  n  (different)  values  «!,  aj,  «3, ,  «n» 

/(a;)"  —  A(x~  tti)  (x  —  ch^  (x  ~  a-^ {x —-  a„), 

in  which  A  is  independent  of  a:,  and  consequently  is  the  co- 
efficient of  a;"  in /(a:)". 

Cor.    If  /(a-)'*  and  ^  (a-)'"  both  vanish  for  the  same  tn  dif- 
ferent values  of  a;, /(a-)**  is  algebraically  divisible  by  <^(a')'". 


THEORY   OF   DIVISORS. 


59 


-l)H2=o. 


I  polynome 

:0. 


15  3  --  0. 
x''  =  Sx-4:. 

>-l)? 

vanish,  if 

on  substi- 

J  is  the  co- 
ame  7n  dif- 


Examples. 

1.    r''-f-aa:''  + ^^  +  ^  will  vanish  if  2,  or  3,  or  —4  be  sub- 
stituted for  X ;  determine  a,  b,  c. 
The  coefficient  of  the  highest  power  of  a:  is  1  ; 

.\a^-{-ax''-{-hx-{-c  =  (x-2){x-?>){x-^4:) 


x^  —  x''-Ux-\-2i', 


a 


1;  b 


14 


24. 


2.  .T^  +  bx"^  -{-  cx-\-  d  will  vanish  if  —  3,  or  2,  or  5  be  sub- 
stituted for  X ;  determine  its  value  if  3  be  substituted 
for  X. 

The  given  polynome  =  (x  -{-S)(x  —  2)  (:r  —  5)  ; 

.-.  the  required  value  is  (3  +  3)  (3  -  2) (3  -  5)  ^  -  12. 


3. 


ax^  -\-  3  bx"^  +  3  ex  -f-  cl  will  vanish  if  for  x  be  substituted 
—  3,  or  "I",  or  1|-,  but  it  becomes  45  if  for  x  there  be 
substituted  3  ;  determine  the  values  of  a,  b,  c,  d. 

The  coefficient  of  the  highest  power  of  a:  is  a ; 

.  ax^  -\-Sbx'^  -i-3cx-\-d=a  (x  -\-  3)  (x  —  ^-)  (x 

.r.  (3 +  3)  (3-1)  (3 -11) -45; 


li); 


a  =  2. 


.  2:^^  3  ^'a;^  +  3  ex  -f  d  -    2  (.r  +  3)  (a:  -  i)  (a;  -  H) 
%  I  a  =  4^. 


h  —  2  .    f,— 


4.    If  or'  ■\-'px-'  -\-  qx-\-r  vanish  for  37=  a,  or  6,  or  c,  deter- 
mine jp,  q,  and  r  in  terms  of  a,  b,  c. 

x^  '\-px'^  -\-  qx-\'r~  (x  -  a)  (x  —  b)  (x  —  e) 

—  x^  —  (a  -f-  i  +  c)  :r^  -|-  (ab  -\-  be  -\~  ca)  x  -  abc. 

:.p-~~  (a-\~b  ~\-  e)  or  —  %  a, 

q  =^  ab -\- be '\- ca    or  Sai, 

r  =  —  abc  or  —  2  a^c. 


THEORY    OF   DIVISORS. 


6. 


6. 


If  a?  -\-px^  -{-  qx-{-r  vanish  for  x  ~  a,  or  b,  or  c,  deter- 
mine the  polynome  that  will  vanish  for  x  =  b-{-  c, 
or  c  -f-  a,  or  a-\-b. 

Since  x^  -}-px^  -\-  qx-{-r  vanishes  for  a;  ==  a,  or  Z>,  or  6?, 

x^  ~px^  -\-  qx  —  r  will  vanish  for  x  =  ~  a,  or  —  i, 
or  —  c,  and  —  ^?  —  «  -j-  5  -f  c. 

But  the  required  polynome  will  vanish  for 

a:  =  — p  —  a,  or  —  p  —  b,  or  ~p  —  n  ; 

that  is,  for  x-\-p^=  —  a,  or  —  b,  or  —c. 
Hence,  it  is  (x  -j-pf  —■p(x  -{-py  +  q  (x  ~\-p)  —  r 

=  x^  +  2px^  +  (i^^  +  ^)  ^  +/'3'  —  '''• 

The  following  is  the  calculation  in  the  last  reduction. 
(See  page  37.) 

1 


P 
P 
P 


■P 


—  r 


1 
1 

1; 
1 


0  q 

p\  p^-Vq 

2p 


pq  -  r 


In  any  triangle,  the  square  of  the  area  expressed  in 
terms  of  the  lengths  of  the  sides,  is  a  polynome  of 
four  dime  isions ;  and  the  area  of  the  triangle,  the 
lengths  of  whose  sides  are  3,  4,  and  5,  respectively, 
is  6.  Find  the  polynome  expressing  the  square  of 
the  area. 

Let  a,  b,  and  c  be  the  lengths  of  the  sides,  and  A  the 
required  polynome. 

1.  The  area  vanishes  if  any  two  of  the  sides  become  to- 
gether equal  to  the  third  side  ;  hence,  \i  a-\-b  --  <?, 
-4  =  0,  and  consequently  A  is  divisible  by  a-\-b  —  c. 
Similarly  it  is  divisible  hj  h-[-o  ~ct  and  \>j  c-\-a  —  b. 


THEORY    OF    DIVISORS. 


61 


or  c,  deter- 


)r  b,  or  ^, 
ty  or  —  i, 


t  reduction. 


^pressed  in 

olynome  of 

iangle,  the 

spectively, 

square  of 

ind  A  the 

become  to- 
ft +  ^  -~  <?» 
a-}-b  —  c. 

yc+a-b. 


2.    The  area  vanishes  if  the  three  sides  vanish  together; 

hence,  if  a  +  ^  +  ^  =  0,  ^  =  0,  and  consequently  A 

is  divisible  by  a-{-b  -\-  c. 
We  have  thus  found  four  linear  factors,  but  A  is  of  only 

four  dimensions. 

.*.  A  =7n(a-{-b-\-c)(b-^c~a)(c-\-a  —  b){a-{-b~-c), 
in  which  m  is  a  numerical  constant. 


But  r,2  or  36 -m  (3+4+5)  (4+5- 3)  (5+3    4)  (3  M    5) 

—  57om;  .'.171  =  -^. 

The  above  includes  all  the  ways  in  which  the  area  of  a 
triangle  can  vanish,  for  the  vanishing  of  only  one  side  in- 
volves the  equality  of  the  other  two  ;  or,  if  a  =  0,  b  =  c, 
and  therefore  a-{-b  =  c,  which  is  included  in  (1).  If  two 
sides  vanish  simultaneously,  the  three  must  vanish. 

Examples  on  the  Corollary. 

7.  Prove  that  (x  +  1)''^  —  x^'^—2x—l  is  divisible  by  2x^ 

+  3:r'  +  a;. 

Factoring  2:r' +  3 o:^  +  rr,  we  find  it  vanishes  for  x  =  0, 
or  —1,  or  — I".  Substituting  these  values  in  the  first 
polynome,  it  also  vanishes.  But  these  are  different 
values  of  x,  hence  the  truth  of  the  proposition. 

8.  (x -\- y-\-zy —  a^  —  y^  ~  ^  is  divisible   by  (x-^y-^-zf 

—  o?  —  if  —  z^. 

The  last  expression  vanishes  if  a:  =  —  y,  so  also  does  the 
first. 

By  symmetry  they  both  vanish  if  ?/  =  — z  and  if  z~  —  x. 
Hence,  they  are  botji  divisible  by  {x\y'){yA^z){z-\'X^. 
But  this  expression  is  of  three  dimensions,  as  also  is 
the  second  of  the  given  polynomes,  hence  it  is  a 
divisor  of  the  former. 


\ 


62 


THEORY    OF   DIVISORS. 


9.    Prove  that 

[(a  +  hf+  {c  +  df]  (a  -h){c-  d) 

+  W^  +  of  +  («  4-  df]  (h  -  a)  (a  -  d) 

J^-[(b  +  df-{-(c  +  af]{b-d)(c-a) 

is  algebraically  divisible  by 

(a-b)  (c-d)  (h-c)  (a-d)  (b-d)  (c-a)(a+b+c^d), 

and  find  the  quotient. 
Let  a  =  b,  and  the  first  polynome  reduces  to 
[(a  +  cf  +  (a  +  df]  (a  -  c)  (a  -  d) 

-{-[{a^df^{c-{-af]{a-~d)(^o--a) 

which  vanishes,  the  second  complex  term  differing 

from  the  first  only  in  the  sign  of  one  factor,  having 

(c  —  a)  instead  of  (ci  —  c). 
Hence,  the  first  polynome  is  divisible  by  a  —  b,  and 

by  symmetry  it  is  also  divisible  by  a  —  c,  by  a  —  d, 

hy  b  —  c,  hj  b  —  d,  hy  c  —  d. 
Again ,  {a + bf + (<? + df  is  divisible  by  {a -f  l>)  -j-  (c -\-  d) ; 

for,  on  putting  a  -f  5  =  —  (c  +  c?),  it  becomes 

[-{c  +  d)Y-\-{c  +  df  =  0. 
Similarly  the  other  terms  of  the  first  of  the  given  poly- 

nomes  are    each   divisible   by  a -\- h  -\-  c  -\-  d,  and 

consequently  the  whole  is  so  divisible. 
Now  all  these  factors  are  different  from  each  other, 

hence  the  first  of  the  given  polynomes  is  divisible 

by  the  product  of  these  factors ;  that  is,  by  the  sec- 
ond of  the  given  polynomes. 
Both   of  these   polynomes   are   of  seven  dimensions, 

hence  their  quotient  must  be  a  number  the  same  for 

all  values  of  a,  5,  c,  d. 
Put  a  =  2,  Z»:=l,  c  =  0,  d  =  -l,  and    divide.      The 

quotient  will  be  found  to  be  —  5. 

:.\{a-\-bf^{c-\-df](,a-b){c-d) 

^[{h-\-cf-{-{a^-df]{b-c){a^d) 

+  [(^^  +  ^)H  (^  +  af]  (h  ~d)(c  -  -  a)  = 


of 


THEORY    OF    DIVISORS. 


to 


Tin  differing 
Lctor,  having 

T  a  —  h,  and 
c,  by  a  —  d, 

omes 

given  poly- 
c  -\-  d,   and 

each  other, 
is  divisible 
by  the  sec- 
dimensions, 
he  same  for 

nde.      The 


Note.    It  is  not  always  necessary  to  find  the  factors  of  the  divisor, 
as  the  following  examples  show. 

10.    Prove  that  x^  ~\-  x-^-l  is  a  factor  of  a;'*  -{-x''  -\-l. 

x^  -f  X  '\- 1  will  be  a  factor  of  :r'*  +  x^  j-  1,  provided 

^.H  _(_  ^^  ^j-  1  =  0,  if  x"  +  a;  +  1  =  0. 
Ifa;^-f-:p+l-=0, 
:.x^  -\-x^  -\--x  =  ^, 

:.:^-Vx''\-x-\-\^\, 
.•.  a:"  =  1  and  x^"^  —  1, 


x^  ^=  X  and  x^^  =^  x'\ 


.'.  x'*-{-x''+l  =  x''-{-x-{-l  =  0, 

.•.  x"^  +  X  -|  - 1  is  a  factor  of  x^*  +  x'^  +  1- 

Two  other  methods  of  proving  this  proposi^'on  are  worthy 
of  notice. 

I.    x^  -\-  x-\-l  will  be  a  factor  of  x^^  }-  x''  -{-  1,  provided 
it  is  a  factor  of 
[(x''  +  x''  +  1)  rh  a  multiple  of  (a;'  +  .r  +  1)]. 

x^*  ~\-  x^  -|- 1  differs  by  a  multiple  of  x!^  -{-  x  -f  1  from 

x''  +  x''  {x'  +  X  +  1)  H-  x^  {x""  +  X  ^- 1)  +  .7;^ 

+  x'  {,v'-\^x\  - 1)  +  .r  (.r^  +  .r  +  1)  H- 1 
=  .r^'^ {x''  +  .r  -1- 1)  ,  x^  (:x'  -[-  .r  -|- 1)  -|-  .^•"(•^•'  "h  0:  +  1) 

+  :r'  {x"  +  x-\-l)  H-  (.r'  -h  -^  -f-  1) 

=  (.T^2  _^_  ^.9  _f_  _^.0  _J_  ^.3  _j_  1-)  (^,^.2  ^_  ^  ^p  i)_       . 

Hence,  x'^  +  •'^'  "h  1  is  a  factor  of  .x''*  -f  .r^  H- 1. 


IT 


.t"  -h  :?.•' 


z+Z'-k-FcZ). 


x^  +  ^'  +  1        a;^  —  1       .1' 
^  jx''  -  1)  r(rg^''  -  1)  -  x{x''  -  1)1 
(:r^ -1)  (.!•=* -1) 

=  (^"'  -  1)  (^''  -  1^  --  .r(.T-'^-l)Gr'^-l) 
(.r^  -  1)  (.r'  -  1)  '       (>  --  1)  (^^^^  -  1)  * 


(VI 


TlIKOllY    OF    lUVISORS. 


Bui  we  Koo  lit  once  tluii  on   riMluetion   l)otli  of  th»>He 
iVactions  give  an  integral  ([uotient ;  lienre, 

(.r^* -}- x''  |- 1)  :- .r' 4- •'^' +  1  give.s  an  integral  (luotient. 

1 1 .    .r"  \  X  I   1  is  a  factor  of  (.7;  - 1-  1  )^  -   .r^      1 . 

J f  .r» -I  .r  I   1  -r  0,  (x  \\y      .7^  -  1  will  vaiUHli  alno ; 
i'or  in  .such  case  .i'+  1  -  -  -    x'\ 


.-.(.rf  IV      x'  ~  I  =^  {-- x') 


.'AT  _  ^.7         1   — 


1^~X'* 


X' 


1, 


which  by  the  last  example  vanislies  if.r'^  |  .r  j-1     0. 
.'.  .r'  -f  X  -|- 1  is  a  f\\ctor  of  (x  -j-  ly     .<:'      1. 


X 


For  ;i:  substitute  *_,  and  multi])ly  by  v/  and  y',  respec- 

lively,  and  this  exanij)le  becomes 
.r"  +  .r?/  -f-  ?/  is  a  factor  of  (x  [-  'f/V  —  a:"'  -  y'. 


„ 


Ex.  21. 

Determine  the  values  of  a,  h,  c,  d,  c,  in  the  following 
rases : 

1.  .r"  +  3/>.r*  -f-  ^cx  -\-  d  vanishes  for  x  —  2,  or  3,  or  4. 

2.  .r*  -|-  r.r'  +  (7.r  -f-  ^  vanishes  for  x  —  1^,  or  --  3,  or  4^. 

3.  x^  -\-  hx'^  +  <^'^*  +  24  vanishes  for  x  ~.  2,  or  —  3. 

4.  <fx^  -\-  hx'^  +  ^'^'  +  90  vanislies  for  x  ==  3,  or  —  5,  or  2. 

5.  ((X*  ~\-  cx^  --  30.r  -f  c  vanislies  for  x  —  1^,  or  —  4,  or  2|. 

6.  81  .?•*  I  -  0  r.r^  -1-  4  dx  -f  ^  vanishes  for  x  =  1 1,  or  --  3-J^,  or  1  J. 

7.  ax*  -f  ^-^'^  +  ("'^^  —  81  vanishes  for  x--  f,  or  J,  or  3. 

8.  (7.r*+  <M''^  f  ^.r  +  ('  vanishes  for  x  —  2,  or  1|-,  or  —1,  and 

becomes  14  for  x  -=  1. 

9.  ax^  +  ex  4- 1?  vanishes  for  x  —  1  J,  or  2|,  and  becomes  49 

for  .r  —  3,  determine  its  value  for  x  —  ~3. 


THKOUY    OK    DIVIHOUS. 


(if) 


vaiiiHli  also 


Ciivon  thai  .r""'     px?  \  qx      r  vanislicH  for  x    -a,  or  A,  or  r, 
(luiormino  ihn  polyjioiiKi  iluil  vani.sho.s  lor  : 


n  -|-  1,  or  />   I   1 ,  or  r  (  1. 
((.       1,  or  h       1 ,  or  c       1. 


10.  X 

11.  X 

12.  .?; 

13.  .r      ((f>,  or  /vr;,  or  ra. 

14.  X    ■  a^,  or  />',  or  r\ 

15.  .f         (^'  |-c),  or  h{c  \  (t),  or  c(a -]- h). 


1       1         1       1         1       1 

1      -,  or  I      -,  or  1 

(>.  h  a 


[ 


f.(J)    \-c)       q 


a 


16.    0. 


^*"H^    «.,  ^''  I  ^'    ,>«  ^'"t  '''''     fa\h      J)       A 


-J  or    -^ — ,  or 
c  a 


,, 


)■ 


becomes  49 


Prove  that  the  following  are  cases  of  exact  division  : 

17.  {x  ~  Xf  -  x''-\-{x'  -  X  -h  \f  -  -  ar*   -  2a;^  +  2a;      1. 

18.  {x  -  1)*«  -  x^ -f  (a;'  -  x-^-Xf-^-x'  -  2a;'  +  2a: 


1 


F.\5 


19.    (.7;  ~  2)'"(2a;  ^  5)^"  -  a;'"  -|-  2'»(a;'  -  4a;  +  5) 


ar*  — 6  a;M^  13  :i'- 10. 


5a;-3-^a;='  +  6a;'+8a;+3. 


20.  (.^'  +  4a;  +  3y»-a;^''-a;' 

21.  (9a;-4f  (a;-iy^-a;^'-(9a;'  --14a;  +  4y' 

-^  (a;  -  l)(9a;- 4)(9a;' -  14a;  +  4). 

22.  [6(a; -  1)]^^  -  (2a;'  +  3a;-  4)"4-(2a;'  -  3a;  +  2)^' 

-^  (2 a;'  -j-  3 a;  -  4)  (2a,-'  -  3 .^  f  2)  (a;  -  1). 

23.  [2(a;+l)(a;-2)]"+(a;'-3.T+3)"-    (3a;'-5a;-l) 

-^  (a;  +  1)  (a;  -  2)  (a;'  -  3 a;  +  3)  (3a;'  -  5 a;  -  1). 

24.  [6(a;-  V)Y  -  (2a;'  +  3a;  -  4)'"-  (2a;'  -  3a;  +  2y« 

+  2(2a;'  +  3a;  -  4)«  (2a,-'  -  3a,-  +  2)* 
-i-(a;~l)(2a;'  +  3a;-4)(2a;'-3a;  +  2). 


m 


TIIKORY    01''    DIVISORS. 


M 


25.  [2(.r -I  l)(.t;      2)]«'-(.r'-3.i--f-3)'^"     (3ar'-5a;-l) 

-\-2(x'  -  Sx  +  3)«  i^x'      bx  -  1)" 
-i-(x\-l)(x~2)(x'~3x\-^){^x'   -bx-1), 

26.  1   j-x^^l-x^-^l-l-x-l-x'. 

27.  ..•'"-|-;^7/^  +  y»-:  .7;'^-f.7;y  f  y^ 

28.  1  4-  r'  4-  .x-«  - 1-  .r"  f  A-'''  :   1   f-  x  -f  .x-'^  }  x'  |  .r*. 

29.  1  -I-  x'  -h  :t'"  H-  x''  -\~x'''-i-l-^x-{-  x'  -f  .r'  (  ;/;*. 

30.  x'^  +  a;^y  -f-  r^?/>"  +  ?/"  --  r"*  +  :r'^y  +  xi/''  -f  y\ 

31.  .-r"  +  a;*  +  r'  -f-  ^  +  1  --  .^''  +  ^'  "h  ^'  +  ^^'  +  1. 

32.  l-l-x  +  x'-l-x^  +  x^-l-x'^i-af^ 

-^.^  1  ~\- X -{- x^ -{- x^ -\- X* -}- r' -^  x^. 

Find  the  quotient  of  the  following  divisions,  in  wliicli  I) 
denotes  the  product : 

(h  ~c){c~  a)(a  -  /;)  (a  -  d)(b  -  d)(c  -  d). 

33.  {h''c^-^a^d''){h-c){a-d)^{c^a:'-\-h''d''){c~a){h-  d) 

+  {a?  IP  +  c^  d')  {a  -b)(c-  d)  ^  D. 

34.  {hc-\-ad)(JP---c^){d'~d'')-\-{ca^hd){c^~a^){h'~d'') 

35.  (5+(?)(a+(i)(52— c^)(a'  — c?')  -|-  the  two  similar  terms 

36.  (Z»^  +  c^)  {a!-  +  c/^)  (/;  -  c)  {a  -  d) -\-  the   two   similar 

terms  -^  2). 

37.  ^Jjc  {b  +  c)^  +  ad  (a  +  J)']  (^^  "  ^')  («  -  t?)  +  the  two 

similar  terms  -^  D. 

38.  [be  (J)  +  c)  +  ac?(a  +  d)]  (b'  -  c^)  (a'^  -  d')  +  the  two 

similar  terms  ~-  D. 

39.  [bc(b'  +  c')  -\-ad(a' -\-  d')](b  -  c)(a  -  d)  +  the  two 

similar  terms  -^  Z). 


1. 


2. 


'I'llKoliY    (>K    M  VISORS. 


\U 


X  -  1). 


m 


x\ 


x\ 


r 
1. 


,  in  which  D 

d). 

'-a)(b-  d) 

-a'){b'-'d') 
imilar  terms 
two   similar 
+  the  two 
-f  the  two 
+  the  two 


40.  (/>-|  ^'      (0      (/)*(/>'-    r)  (a— (Z)  I  •  the  two  siiuihir  terms 

i-  A 

41.  Tiic  sum  of  the  friictions  -J-,  -J,  J-,  >»-,  inennised  ])y 

the  sum  of  their  products,  two  hy  two,  incroascMl  ])y 

the  sum  of  tlicir  products,  throe  ])y  thn^e,  ,  iii- 

croasod  ])y  their  product  is  erpial  to  7i. 

42.  In  any  trapezium,  llie  H(piiire  of  tlio  area  expressed  in 

terms  of  tlie  h'li^tlis  of  tlio  paraUcl  si(h\s  and  tlu^ 
diagonals,  is  ;i  polynome  of  four  dimensions;  deter- 
mine that  polynome. 

43.  In  any  (pi ad ri lateral  inscrihed  in  a  circle,  the  square 

of  the  area,  expressed  in  terms  of  the  lengths  of  the 
sides,  is  a  polynome  of  four  dimensions ;  find  that 
polynome. 

Tfieorp:m  ITT.  If  the  polynome  f(xY  vanish  for  more 
than  n  different  values  of  x,  it  vanishes  identically,  the  co- 
cfTicient  of  every  term  being  zero. 

Cor.  If  a  rational  integral  expression  of  n  dimensions  be 
divisible  by  more  than  n  linear  factors,  the  expression  is 
identically  zero. 

Examples. 


1. 


{x  —  a)  (x  —  b)  (x  -  h)  (^  "  ^)  I  (•^"  -  ^)  (^"  -  ^0  1  A 
{7^f{^^)  '^  (V=  6)7^^^  +  (/,  _  c)  (b  -  a) ~^^^' 
if  a,  b,  and  c  are  unequal ;  for  this  is  a  polynome 
of  tivo  dimensions  in  x,  but  it  vanishes  for  x  ~"  a, 
and,  therefore,  by  symmetry,  for  x  =  b,  and  for  a;—  c; 
that  is,  for  thj'ee  different  values  of  x ;  hence,  it  van- 
ishes identically. 

2.    [(a-\-?^y  +  (c^'dy]ia-'.)(c-d) 

+  [(^  +  cy+(^-^  +  ciy]  (b  ~  c)  (a  -  d) 

+  [(^  +  «)'+  (P  +  dy]{c-a){b-d)  =  0. 


()H 


TIIKoUY    (»l'    ItlVISOllS. 


SiiljHtilut*;  h  lor  ii^  jiiid  IIk!  cxprciHKioii  bcujomiiS 

-\-[{n-ihy-\-{h^\-dy]{c-h){h-d\ 

which  vaiiiHhes;  hoiicc,  tho  givon  oxpreHsion  is  divi- 
sibh;  ])y  a  h,  and  conseijuently,  l)y  Hymmotry,  it  is 
divisible  l)y  ((t~  h),  (h-c),  (c-  d),  (a  —  c),  {J>-  d), 
jind(a  -</).  But  the  given  expression  is  of  only 
four  dimensions,  while  it  a[)pears  to  have  six  linear 
factors  ;  hence,  it  vanishes  identically. 


Ex.  22. 


Verify  the  following; 


':i  .,2 


x^  -f  y'  +  z'  - 1)"  -  c" 


4- 


C\(j'~b'') 


fz'  ,   {f  -  b^)  jz'  ~-  P)   ,   0/    -  cQ  (z'  -  r-O 


b 


'/c' 


4- 


b\b''-c') 


+ 


C'^(6''^~Z»'0 


x'y'      {x''-^ce){a^~y')z''  ,   (x' ~b')  (b' -■  y')  z 


a 


2  A2 


+ 


+ 


6'^      (z^-a^)(6'^-a'0a'      (b''-z')(b''-a')b' 


(z^-X^)(z^~?/)_^r. 

^  (z'  -  d')  (b'  -  z') 


4. 


+ 


(x  4-  «)  (a  •—  b)  (a  —  ^)      (a;  -f  b)  (J)  —  r)  (/6  -  -  a) 
1  1 


+ 


{x  -\-  c){c  —  a)  (o  ~  b)      (x  +  a)  (x  +  b)  (x  +  c) 

5.    bc(b'-c')-}-ca(r--a'')  +  ab(a'~b') 

=  (a  +  bi-  c)  [a\b  -  c')+  Z'^(c  -  a)  +  c'ia-b)]. 


6. 


+ 


^(^— y)  (^-2;)    y  (y  -  ^)  (y  -  2)    2(2  -  x)  {z  —  y) 

a 


xyz 


THEORY    OF    DTVTSORK. 


m 


lU'H 


iHflion  is  (livi- 
rumotry,  it  ih 

»n  is  of  only 
vu  six  linear 


1. 


i\»i 


6'^ 


a) 
b)  {x  +  c)' 


^)  (2;  -  y) 


•        tV'  (6  -    c)  +  //^  (c  -  a)  -f  6'^  {it  -    /;)  " 

8.    {<i(1f'\-  h(;f-\-  bed  -~  anf  \   (bee  \-  acd  +  ,irj'-     Inljf 


9. 


(«  —  b)  {b  —  c)  {c  -  a) 

=  i[{ct~by^{b-cf~[-{c-ay\ 

10.    (     a---fy  +  2)(a;  — 2/  +  2)(^-}-y  — z) 

^\-'x{.v-   y-\-z){x-{-y  —  z)-\-y{x-\-y-  z){  -X'\-y-\'z) 
'h  z(-  x-\-y  -f-  2)  (^'  -  y  -[■  z)      \xyz. 

11     U^'^>y-f(^>'-6'y +  (<•'- ^^-7 
(«  +  />)  ("^  -f  6')  (6'  -h  a) 
---.  (^t  -  i)^  +  (^^  -  cf  +  (6-  -  «)'. 

12.    x\y  -1-  s)'^  -F  y\z  +  ^f  +  z'^(./;  ^-yf  -|-  2.7y2(.i'  -j-  y  -f  2) 
:-  2  (.ry  +  yz  -\-  zx)\ 

TiiEUREM  IV.  If  the  polynomes  /(.f)'*»  ^{^Y  (^  '»***' 
less  than  m),  are  equal  for  more  than  71  different  values  of 
.r,  they  are  equal  for  all  values,  and  the  coefficients  of  e(|ual 
powers  of  x  in  each  are  equal  to  one  another. 

This  is  called  the  Principle  of  Indeterminate  Ooefficients. 


1. 


a' 


Examples. 


+ 


Jy" 


{a  —  b)  {a  —  c)  {a  —  d)      {b      a)  {b  -  -  c)  {b  -  d) 


-h 


+ 


d' 


{c  -  -  a)  {c  —  b)  {c  —  d)      {d—  a)  {d— b)  (d —  c) 
Assume 


-=0. 


X 


(x  —  a)  (x  —  •  •  (x  —  c)  (x  —  d) 


X  —  a     x  —  b  '  x  —  c  '  X  —  d 
in  which  A,  B,  C,  D,  are  independent  of  x. 


(«) 


I- 


70 


THEORY    OF    DIVISORS. 


Multiply  by  (x  —  a)  (x  ~b){x~  c)  (x      d). 

.'.  x^  =  (A~{~  B  -\-  C-j-  D)x^  +  terms  in  lower  powers 


of 


X. 


Now  this  equiility  holds  for  more  than  three  values  of 
X,  holding  in  fact  for  all  finite  values  oi  x. 


Again,  multiply  both  sides  of  (a)  ])y  x  —  a, 


m 


(x  —  b)  (x  —  c)  (x  —  d) 


=  A  + 


\X  — 


T  + 


c 


^^  --- 


Z) 


•//      ~  \J  *AJ 


X      a] 


Put 


X  =:  a. 


a 


(a  —  b)  (a  —  c)  (a  —  d) 


A. 


By  symmetry, 
Adding, 


{b-a){J)-c){h~d) 


--  B,  etc. 


a' 


^ 


(a-b){a-~c){a-d)      (b  -  a){b  -  c){b  -  d) 


+ 


+ 


d' 


{c  —  a)(c  ~  b)(c—  d)      (d—  a) (d  —  b) (d  —  c) 


2. 


a' 


=  ^  +  ^+(7+D=-0,  by(/3). 

{a-\-b){a-\-c)      b\b-\~c){b-\-a)  .  c\c-^a){c-\-b) 


+ 


4- 


(a-b)(a  —  c)         {b  —  c)(b~a)         (c  —  a)(o  —  b) 
=  (a  +  5  +  c)\ 

Assume  x^  — -  px"^  +  ^-^  —  r  =  (x~a)  (x — b)  (x — c)     (a) 

.-.  a^  -i-2)x'^  +  qx  -^  r  —  (x -{- a)  {x  ~j-  b)  {x  +  a)  (fS) 

x*-{-px^-{-qx^-\-rx  .  ^      .     A     .     B     .      C     ^  . 

3r — px  -f- qx  —  r  x—a      x—b      x—c 


THEORY    OF    DIVISORS. 


lower  powers 
iree  values  o^ 

)f  X. 

a. 


a 


)• 


B,  etc. 


-b){d-c) 


0  —  a)  (c  —  b) 
>)(x-c)     (a) 

It  +  -"  (y) 


Multiply  by  x^  ~ px^  -{-  qx  —  r,  and  equate  tlie  coefii- 
cients  of  the  terms  in  x"^.  In  multiplying  the  frac- 
tions in  the  right-hand  member  of  (y),  use  the  factor 
side  of  (a). 

.\A  +  B-[-C==2p\ 

Multiply  both  members  of  (y)  by  x      a. 

X  (x  -f-  g)  (x  -{-  h)  (x  -j-  6') 
(x--b)(x  —  c) 

Put  a;  =  «, 

2a'(a-i-b)(a  +  c) 
(a  —  b)  (a  —  c) 

By  symmetry, 

2b'(b-{^c)(b  +  a)  ^  ^^  ^^^^^  2c'(c  +  a)(e-\-b)  ^  ^  ^^ 
(b  —  c)  {b  —  a)  \  {c  ~a){c  —  b) 

■  a'(a  +  h)(a-\-e)      V {b -\- c) (b -\- a) 
"    {a-b){a-c)    ~^    {b-~c){b-a) 

((?  ~a){c  —  b) 
=  {a-\-b-\-c)\ 

3.    Extract  the  square  root  of  1  -\-  x  -\-  x^  -{- x^  '\-  x^  -[■ 

Assume  the  square  root  to  bo 

l-^ax-{-bx''-{-  cx^  -f  dx'  + 

.-.  l-\-x-\-x'-{-x^-\-x'-\- 

=  (1  +  ax  +  bx^  +  car*  +  dx^  -f )' 

=  1  +  2ax  -f  (a'-f-  2b) x^  -f  2(«5  +  c)x^ 

+  {2d-\'2ac^h-')x'-\- ^ 


J 


7i2 


THEORY    OF    DIVTROR.^;. 


2a -=1, 

2(c  +  ah)  =  l, 
2d+2ac-i-b'=l, 

V(i+^  +  ^"'  + ) 


a  — 


J' 


^  ~i~(i  ^  I) 

I      3  5    ^4    I    


3 

¥' 

_5_ 

—  16» 

9  ^  — 


35 
ITT- 


Note.  As  it  is  frequently  necessary  to  determine  the  coefficient  of 
ii  particular  power  of  x,  a  few  preliminary  exercises  are  given  on  this 
subject. 

Ex.  23. 

Determine  the  coefficient  of: 

1.  x'  in  (1  +  axf  +  (1  +  bxf  +  (1  -  cxf. 

2.  :i^  in  (1  +  X  +  2ii^  +  3r^)  (1  -  x  +  S^r^  +  a;-' -  5:?;^). 

3.  x'  in  (1  +  X  -\~  2x'    -  3:r'  +  ix*  -f ) 

4.  x^  in  ^(a;  — Z>)(a;—  6')(a;  —  d) -\- B (jv  —  a) (x  —  c) (x  —  d) 

+  ^0^'  -  -  a)  (:r  —  Z»)  (a;  —  d)-[-D (x  —  a)(x  —  b)  {x  —  6*) 

5.  .^•*  in  (1  —  axY  (1  +  axf. 

6.  x^  in  (1  4-  axy{l  —  bxf. 

7.  In  the  product 

(1  -\-  ax  -\-  bx^  -\-  ca?  -{- )  (1  —  ax  -\-  bx^  —  cx^  -f ) 

prove  that  the  coefficients  of  the  odd  powers  of  x 
must  be  all  zeros. 

Determine  the  value  of  the  following  exjiressions : 

8  1  ,    ^ 

*    (ii  ~b){a-  c){a  ~d)      {b  -  a){b  -  c){b-  d) 

1  .  1 


+ 


+ 


{c-a){c-b){c-d)      (d-a)(d-b)(d-c) 


THEORY    OF    DIVISORS. 


73 


)- 

i. 

f) 

5 
TS"' 

S"  " 

A)  = 

1 '2  li- 

5   ^3 

the 

i  coefficient  of 

arc 

given 

on  this 

or'—  5  a;*). 


i)(x  —  b)(x—c) 


hx'-cx'i- ) 

dd  powers  of  x 

rensions : 

I 

'-h){d-c) 


9. 


10. 


11. 


12. 


13. 


a 


+ 


a~h)ia-c){a-d)      {b  -  a){b  -  c){b  ~  d) 


+ 


a' 


a  —  h){a—c)  (a  —  d) 


a 


a  —  h)  {a  —  c)  (a  —  d) 


-f  three  similar  terms. 


-f  three  similar  terms. 


a* 


a  —  h)(a  —  c)  {a  ~-  d) 
bed 


a  —  b)  {a  —  c)(a  —  d) 
14^    a(a-\-b){a-\-c) 


-f  three  similar  terms. 


-f  three  similar  terms. 


(a  —  b){a~-  c) 

-_     a^{a-{-b){a-{-c) 
{a  —  h)  {a  --  c) 


-{-  two  similar  terms. 


+  two  similar  terms. 


-       a\a-{-b){a-\-c)   .    ,         •    -i      , 

lb.      -^ —  ;^<^ /  -\-  two  similar  terms. 

{a  —  b){a  —  c) 

-^     a(a-{-b)(a-\-c)(a-{- d)  ,  ,,  ••14. 

17.  ^. — '--^>- — ! — ^-x — LA  ^  three  similar  terms. 

{a  —  bjia  —  c)  {a  —  a  ) 

18.  --> — -^/S     '"   ;^    "'    -/  -{-  three  similar  terms. 

(a  —  6)  (a  —  ^)  (a  —  a) 

19.  «'(!l+(')(«±_^)('!±f )  +  tlu-ee  similar  terms. 

{a  -  6)  (a  —  c)  (a  —  d) 

20. ^— - — I—-  -(-  two  similar  terras. 

{a  ~b){a~c) 

For  numerator  use  px^  —p^x  -\-pq  —  r, 

^1.    i_^ ^ L^  _L  two  Similar  terms. 

{a  --b){a  —  c) 

For  numerator  use  ?fx^  -\-px-\r  q. 


•    '■s^!^^-\. 


74 


22. 


23. 
24. 
25. 


olb  fc) 


THEORY    OP    DIVISORS. 


-T  -{-  two  similar  terms. 


(a  ~b){a  —  c) 

For  numerator  use  a:(:?;  — />). 


(((,  —  b){a-  c)  (a  —  c/) 


+  three  similar  terms. 


a^  (he -{- cd  4- dh)        ,   .-i  •    -i      , 
^_y -'— — /--|_  three  similar  terms. 

{a  —  0)  [((,  —  c)  {a  —  a) 


he  +  cd  +  db 


-j-  three  similar  terms. 


(a  —  />*)  (a  —  c)  (a  ~  d) 

Extract  the  square  root  to  four  terms : 
26.    l  +  .r.  27.   l-x.         28.   \-\-2x-\-2>x'^^x'^ 

29.  1  -  4.r  r  10.^^''  -  20 a;^'  +  35.^'* -  56:^^  +  84a;«. 

30.  Extract  the  cube  root  of  1  +  ^'  to  four  terms. 


§  11.  1.  Find  the  condition  that  px^  -]-  2qx  -\-  r  and 
p^x^  -\-  2(1  £  -|-  r'  shall  have  a  common  factor. 

Multiply  the  polynomials  by  p^  and  p  respectively,  and 
take  the  difference  of  the  products ;  also,  by  ?•'  and  ?-,  re- 
sj)ectively,  and  divide  the  difference  of  the  products  by  x. 


p'px^  +  2p^qx  -\-p^r 


2 (pq'   -p'q) x-\-(2Jr'—-p'7-) 


pr'x"^  _j_  2  qr'x  -\-  rr* 
p^rx^  +  2  q^rx  -f  tW 


(pr^  —p'r)  a; + 2  (qr'  ~  r'q) 


Multiply  the  former  of  these  remainders  by  (pr'—p'r),  and 
the  latter  by  2(pq'—p*q),  and  the  difference  of  the  products 


IS 


(j^r'  —  pWy  ■-  4:(pq'  —p'q)  (qr'  ~  r'q). 

But  if  the  given  polynomials  have  a  linear  factor,  this 
remainder  must  vanish,  or 

(pr'    -pV)'^  -  4  (p^'     p'q)  (qr* —  r'q). 


THF.OTIV    OV    DIVIt-'OUS. 


75 


lear  factor,  tins 


If  the   given  polynomeB  L/,ve  £    quadratic  factor,    the 
linear  remainders  must  vanisl  idcutioaliy,  or  (Th.  III.), 


or, 


£  =  1 


,.f 


2.  Find  the  condition  that  px^  -f  3  qx^  -\~  3  r.r  -f-  s  shall  have 
a  square  factor. 

Assume  the  square  factor  to  be  {x  —  af.  On  division, 
the  remainder  must  be  zero  for  every  finite  value  of  x,  and 
consequently  (Th.  III.)  the  coefficient  of  each  term  of  the 
remainder  must  be  zero.  Divide  by  {x  —  a)^  neglecting 
the  first  remainder. 


a 

P 

3q                   Sr 
pa         pa"^  -{-  Sqa 

8 

a 

P 

pa  +  3  <7         pa^  +  3  qa  +  37*; 
pa                 2^i/«^  +  3  qa 

E 

P 

2 pa  +  F>  -• ;  3 : pa^  -\-  2qa-}- ')) 

:.pc^  +  2  <7a  -|-  r  =  0 : 

.•. px^  -|-  2qxArT\^  divisi' >^-  by  a;  —  a  ;  (Th.  I.  Cor.  2.) 
or,  py?  -f  3  qx^  -|-  3  r:i:  -f  .s  and  px^  +  2  qx  4-  r  have  a  common 
divisor.  Multiply  the  latter  polynome  by  a?,  and  subtract 
the  product  from  the  former,  and  the  proposition  reduces  to 

\i px"-{'^qo(?-\-^rx-\-8  have  a  square  factor,  px?-\-2qx-\-r 
and  qx^-\-2rx-\-8  will  have  the  square  root  of  that  factor 
for  a  common  divisor. 

Ex,  24. 

1.    Determine  the  condition  iio"(^ssary  in  ord«^i'  that 

x}-\-px-\-q  and    :i^  H-^>'a;  +  </'   may  have  a  common 
divisor. 


70 


THEORY    OF    DIVISORS. 


2.    The  expression  a;«+ 3a V+3i:r*  +  ca;H3c/^•'  +  3c^r+/•'' 
will  be  a  complete  cube  if 
re      d     c~  a^      7        4 


e_ 
a 


h        6a^ 

3.  Prove  that  aoi^'\-hx  -\-  c  and  a-\-hx*  -f  cx^  will  have  a 

common  quadratic  factor  if 

l-^  c'  =  (c2  _  a^  -I-  h")  {c"  -  a^  +  ah). 

4.  Prove  that  ax^ -\- ha^ -{- c  and  a-{-hoi^-{-cx^  will  have  a 

common  quadratic  factor  if 


a 


'11.1 


h^  =  (a'  _  c')  (a'  -  c'  +  he). 


5 .  Prove  that  ax* + ir'' -f  <?^  +  <^  and  a-{-hx-{-  cx^-\-  dx*  will 

have  a  common  quadratic  factor  if 

{a  +  d)(a-dy  =  (b-c)(hd-ac). 

6.  a;'+i5a;^  + 2'a:  +  r  will  be  divisible  by  x"^ -\- ax -{- h  if 

a?  —  2pd^  -{-{p^  -{-  g)ci-\-  r  —  2^q  =  0, 
and  h^  —  qh"^  -\-  rph  —  r^  =  0. 

7.  x"^  -\~px  +  2'  will  be  divisible  by  x^  -\-  ax  -f-  5  if 

a«-4^a'^=/and  (b' +  q)  (h' -  qf -^  p' h\ 

Determine  the  condition  necessary  in  order  that : 

8.  x*-\-4:px^-\-6qx^-{-4:rx-\-t  may  have  a  square  factor. 

9.  ax*  +  4  hx^  +  6  ^"^'^  +  4  cZo;-}-  e  may  have  a  complete  cube 

as  factor. 

10.    x^-{-10hx^ -\- 10  cx^ -\-bdx-{- he  may  have  a  complete 
cube  as  factor. 


3c/a,•H3c^x+/ 


ex"  will  liave  a 


ex"  will  have  a 


r:^cx^-\-dx^  will 


X 


■\-  ax^h  if 


+  5  if 

er  that : 
pare  factor, 
complete  cube 

ive  a  complete 


CHAPTER  III. 

DiRlJCT  ArrLICATION    OF    THE    FUNDAMENTAL  FORMULAS. 

Formulas  [1]  and  [2].     {x-izyy~x^±.^xy-\-y\  etc. 

§  12.  From  this  it  appears  that  a  trinomial  of  which  the 
extremes  are  S([uares,  is  itself  a  square  if  four  times  the 
product  of  the  extremes  is  equal  to  the  square  of  the  mean, 
and  that,  to  factor  such  a  trinomial,  we  have  sim})ly  to  con- 
nect the  square  root  of  each  of  the  squares  by  the  sign  of 
the  other  term,  and  write  the  result  twice  as  a  factor. 

Examples. 
1.    4a;*  -"-  my^y"  f  400?/*  =  (2a;'^  -  20?/^)  (2.^•2  -  %)if). 

3.  {a~hy-\\h      cy^2{a-^h){h      c). 
This  equals 

(^a  ~h  -^h  —  c^{ci  —  h  -\-h  --  c)^  {a~  c){a—  c). 

4.  x^  +  2/^  +  2^  +  2.r?/  —  2,xz  —  2yz. 

Here  the  three  squares  and  the  three  double  products 
suggest  that  the  expression  is  the  square  of  a  linear 
trinomial  in  x,  y,  z. 

An  inspection  of  the  signs  of  the  double  products  ena- 
bles us  to  determine  the  signs  which  are  to  connect 
X,  ?/,  z ;  we  see  that 

1.  The  signs  of  x  and  ?/  must  be  cdihe ; 

2.  The  signs  of  x  and  z  must  be  different ; 

3.  The  signs  of  y  and  ;:  must  be  different. 

Hence,  we  have  x-{-y—z,  or  —  a: -?/  |-z  — -- (.r-f-y -2), 
and  the  Victors  are  (x  -\-y  —  -)  ix  +  y  —  z). 


'8 


FACTORING. 


Ex.  25. 

1.  9 w'  + 12771  +  4;  c2"*-2c'"+l. 

2.  ?/'-2?/2''  +  2'';  lGa;'2/HlGV  +  4/. 


4.    -kTM-lG?/^'-- 4a;^/z 


In* 


'    ¥ 


rt' 


rt'^Z>'V/2+  U'c* 


5.    (a  +  ^>)'^  +  c'  -  2c(a  +  ^') ;  9x'  -  f  .r^/  +  J./ 


6.    2^  +  (;i'-7/y--2z0:--y) 


2m 


f-   - 


a 


'im 


7 .  (.r-^  -  yy  +  2  (:r;^  -  ij)  {y  -  ^'^  +  (y  ~  2^)^ 

8.  (.//^  -  xyY  —  2  (.r^  -  xy)  {xy  -  ?/)  +  {xy  -  y^)l 

9.  (a  +  ^*  +  cy~26'(a+i  +  d^)  +  6-^  Ji./- _ 2/^^  +  J/ (/^ 

10.  (3:r  -  4?/)*'  -h  {2x  -  2>yy  -  2(3a;  -  4y)  (2.^-  -  3y). 

11.  (.x-'^-a:?/  +  2/7+Gr^  +  .Ty  +  y7  +  2(.T*  +  a;'^3/H2/*). 

12.  (52;'^  +  2a;?/+7y7  +  (4.r2  +  Gy7 


2(42:*^  +  Gy'O  (5^^'  +  2a;'2/  +  7 v/^^). 


13. 


'2m 


H- 


a) 


2n 


14.  a'^  +  ^»''  +  c'^-2aZ>-2Z»d?  + 2(2(7. 

15.  a*  +  h'-\-c'~  2a'b'  -  2a^c^  -\-2}?c\ 

16.  {a  -  5)*^  +  {h  -  cy  +  (c  -  a)2  -h  2(a  -  ^) {h  -  c) 

-2{a-h)(c  -  a)  ^2{h  -  c)  (a  -  c). 

17.  4a*  -  I2a'b  +  9^>^  +  IGa^c  +  IGc^  -  2Uc. 


Formula  [4].     x"^ —  y^  =  {x-\-y){x  —  y). 

%  13.  Ill  this  case  we  have  merely  to  trke  the  square  root 
of  each  of  the  squares,  and  connect  the  results  with  the 
sign  -f-  for  one  of  the  factors,  and  with  the  sign  — -  for  the 
other. 


FACTORING. 


79 


-\y\     ' 


^b*c* 


+  V(r2/' 


Koj* 


im 


-  yJ- 

)(2.r-32/). 


Examples. 

1.  {a-\~by~{c-\-d)\ 

Thi.  -  [{a  +  b)  +  {c  +  d)]  [{a  +  Z.)  -  (.  +  d)] 
^={a'\-b-\-c-\-d){a-\-b--c  —  d). 

2.  Factor   {xr  -\-bxy  -\-  y'^y  —  {x^  —  xy  -f-  y'^f. 

Here  we  have  [(:f^  -f-  5 xy  -\-  y'f)  -|-  (ar*  —  rr?/  -f  ?/■)] 

=  2(a:''  +  :r?/  +  ?/2)(G.r?/) 
^\2xy{x-{-y)\ 

3.  rt^-i'^-c'^-f  2Z)c. 

This  =  a'  -  (i  -  c7=  (a  +  Z»  -  c)(a  -  Z>  +  6). 

4.  Resolve   (a' +  i^  -  (a^  -  Z*^  -  (a^ -f  Z/^  -  f 7. 
This  =  4  a^  Z>2  -  (a'^  +  Z.2  _  c'^)^ 

--=  {2ab -ir  o?  +  b^  -  c'^)  (2aZ>  -  a^  -  i^  _^  ,^^^ 
The  former  of  these  factors 

-  (a  +  Z*)'-  c'^  --  (a-1-  b  +  c)(a  +  i  -^); 
i          and  the  latter 

-  &  ~  {a-bJ  =  (c-\-ci~b){c-a  f  b). 
.'.  the  given  expression 

=  (a  -i-  b  -^  c)(a  -\-  b  -  c)(c  +  a  -  b){c  -  a  i-  b). 


I 

Ex. 

26. 

b)(b-c)                 i 

1. 

^da'~4:b\ 

9. 

81  a'      1. 

Abe. 

ft 

2. 
3. 

da'      ^^b\ 
SI  a'      lQb\ 

10. 
11. 

a*      16  Z;*. 
a''     b'\ 

r  -  2/). 

the  square  root 

esults  with  the 

sign  --  for  the 

4. 
5. 
6. 

7. 

100  a,-^      S(j2/. 
bceb      20  bx'y\ 
9x'      16/. 
Ac^-1. 

12. 
13. 
14. 
15. 

a'      b'i^2bc      c\ 
{a-\-2bf      {^x      ^yj 
{x'-^fj      \x'y\ 
{x-\-yy-^z^. 

0 

8. 

4/      |r^^2^ 

16. 

('^x-\-hy      (5a;  +  3)^ 

80 


FACTORING. 


17.  \xhf   ^{x'^'y'-zj. 

18.  {x'-\-xy-y'y~{x'~xy--yy, 

19.  {x'-y'-^-zJ-ix'z'. 

20.  {a-^h~\-c-\-dY~{a~h^-C''  d)\ 

21.  (2  +  3a;  +  ^x^  -  (2  -  Sx  +  4a,-7. 

22.  {a'  +  7/  +  4 ahy  -  (a'  -|-  by. 

23.  (a^-/;^-^c^-(^7-(2ac-2^*(/)^ 

24.  (.f'^-?/'^-s7-  4?/s'\ 

25 .  (a"  -  a^*  z^^'  +  ^")'  -  {a''  -  5  a^  Z»-'  +  bj. 

26.  a''^  -  />''^  +  (ja'b'-  (jb'a'  +  8Z>«a='  -  Sa'b\ 

27 .  (.r^  -h  y^ -\- z^  —  ^y  —  yz  —  zxy  —  (xy  +  yz  +  za:)''' 

28.  (x""  +  ?/''  H-  2'  —  2a;?/  +  2a;z  --  2?/2)  -  (y  +  2)'. 

29 .  2  tt^  //  4-  2  Z/'^  6'"'  +  2  c'  a"  -  a*  -  b*  -  c\ 

30.  .T*  +  7/  +  2'  -  2a.-'/  -  2/2''  -  22'' a;^ 


Formula  [A].     (x-{-r)(x-\-s)  —  x'^-\-{r-{-s)x-\-rs. 

Examples. 
v/  1.    .^•«-9a;'  +  20  =  (.^''-5)(a;^-$■4).'r-u^ 

2.  (a; -  yy  -\-x-y-  110  ■-=  (a; -  y  +  11) (a;  - y  -  10). 

3.  (a'  -  a6  +  Z>7  -j-Qb^a'  -  ab  +  i'O  -4a'  +  9b'' 

=  [(«'  -  a&  4-  h')  +  (2a  +  37>)] 
X  [(a'  -  ai  +  i')  -  (2a  -  3  b)]. 

4.  (.^•'-5:^•)'-^(^''-5^)-40 

=  (a;' -  5.1- +  4)  (ar^ -- 5a;  -  10). 

5 .  {ax  -j-  Z>?/  +  cy  —  ('^^^  —  ^)  («^  +  ^y  +  ^)  —  WW 

=  (aa;  -j- by -}- c  —  7n)  (ax  -{-by-\-c-i-n). 


FAf'T(1RIN<}. 


81 


() 


§  14.  It  will  1x1  soon  thill  tlio  first  (dt  rnin/tmn)  ItTiii  of 
thf!  reqiiirtMl  I'iictorH  is  obt.'iiiiotl  l)y  oxtractiiig  the  stpiaro 
root  of  tli(^  first  torin  of  tho  given  exproHsion,  .'uul  that  the 
thcr  terms  are  {lotorniiiied  by  observing  two  conditions  : 

I.  Their  product  must  equal  the  third  term  of  the  given 
expression. 

II.  Their  ahjchrdif  sum  multiplied  into  the  cnmvton. 
term  already  found  must  equal  the  middle  ivnw  of  the 
given  expression. 

Ilenee,  to  make  a  systematic  search  for  inti^gral  factors 
of  an  expression  of  tlui  form  x'^±:hx±c,  we  may  proceed  as 
follows : 

1.  Write  down  every  pair  of  factors  wliose  product  is  c. 

2.  If  the  sign  })efore  c  is  -|-,  select  the  pair  of  factors 
whose  su7n  is  h,  and  write  both  factors  x-[ ,  if  the  sign  before 
^  is  -f- ;  X'l  if  the  sign  before  h  is  --. 

3.  But  if  the  sign  before  c  is  — ,  select  the  pair  of  factors 
whose  difference  is  /;,  and  write  before  the  hirgcr  factor  x-\- 
or  a;—,  and  before  the  other  factor  x—  or  .6'-|-,  according 
as  the  sign  before  h  is  -f  or  — . 


Examples. 

1.  0;'^+  9;c-f20.     The  factors  of  20  in  pairs  arc  1  and  20, 

2  and  10,  4  and  5.  The  sign  before  20  is  -|-;  hence, 
select  the  factors  whose  sum  is  9.  These  are  4  and 
5.  The  sign  before  9  is  -j-;  hence,  the  required  fac- 
tors are  {x-\-A){x-[-b). 

2.  a;^-8:r+12.     Pairs  of  factors  of  12  are  1  and  12, 

2  and  6,  3  and  4.  Sign  before  12  is  + ;  therefore 
take  the  pair  whose  sum  is  8.  These  are  2  and  6. 
Sign  before  8  is  — ;  hence,  the  factors  are  (^  —  2) 
(:r-6). 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


1.0 


I.I 


■50     ^^~       IIII^H 


lU 

u 


■  4.0 


12.0 


1.25  III  U    |,.6 

M 

6"     

► 

y] 


^ 


/a 


y: 


/A 


'>W 


Hiotographic 

Sciences 
Corporation 


33  WIST  MAIN  STRHT 

WEBSTER,  N.Y.  MStO 

(716)  S72-4503 


^>' 


■^ 


82 


FACTORIN(}. 


3.  x"  -  2i:r-100.     Pairs  oi  factors  of  100  are  1  and  100, 

2  and  50,  4  and  25,  5  and  20,  10  and  10.  Sign  be- 
fore 100  is  — ;  therefore,  take  the  pair  whose  differ- 
ence is  21.  tliese  are  4  and  25.  The  sign  before  21 
is  -  ;  therefore,  x—  goes  before  25,  the  larger  factor, 
and  the  factors  are  {x-{-^)(x~2b). 

4.  .6-^  -I-  I2x  -  108.     Pairs  of  factors  of  108  are  1  and  108, 

2  and  54,  3  and  36,  4  and  27,  0  and  18,  9  and  12. 
8ign  before  108  is  — ;  therefore,  take  the  pair  whose 
difference  is  12.  These  are  G  and  18.  Sign  before 
12  is  -f-;  therefore,  x-\-  goes  before  18,  the  larger 
factor,  and  x—  before  6,  the  other  factor.  Hence, 
the  factors  are  {x  —  6)  (x  -\- 18). 

Note.  It  will  bo  found  convenient  to  write  the  factors  in  two 
columns,  separated  by  a  short  space.  Taking  Exam.  2  above,  pro- 
ceed thus  :  Since  the  sign  of  the  third  term  is  +,  write  the  sign  of 
tlie  second  term  (in  this  case  — )  above  both  columns. 


1  12 

{x-2)       {x-e>) 

Exam.  3  above.  Since  the  sign  of  the  third  term  is  — ,  write  the 
sign  of  the  second  term  (in  this  case  — )  above  tLe  column  of  larger 
factors,  and  the  other  sign  of  the  pair,  ±,  above  the  other  column. 

+ 


5. 


1 

100 

2 

50 

(X 

:-  -)       (^ 

-25) 

x'     34 

a; +64. 

Here 

we  have 

the  factors 

1 

64 

X  - 

-2        X 

-32 

4  16 

And  since  the  last  term  has  the  sign  -f-,  and  the 
middle  term  has  the  sign  — ,  we  write  —  over  both 
columns. 


FATTORIXn. 


83 


irc  1  and  100, 
10.  Sign  be- 
r  whose  ditfer- 
sign  before  21 
)  larger  factor, 

ire  1  and  108, 
18,  9  and  12. 
the  pair  whose 
I.  Sign  before 
18,  the  kirger 
ictor.     Hence, 

le  factors  in  two 
im.  2  abovo,  pro- 
'ritc  tho  sign  of 


6.    2:'-f  12a;  -  G4. 


II  is  — ,  write  the 
;olunm  of  larger 
ther  column. 


-f,  and   the 
—  over  both 


X 


1 

2 
4 


+ 
64 
32 

oj+lG 


H<;rf',  since  the  last  term  has  the  sign  — ,  wo  write  tin; 
sign  (+)  of  the  middle  term  over  the  column  of 
larger  factors,  and  tho  sign    -  over  the  other  column. 

7.    :i-*-10a-^    -144. 

Here  we  have  the  pairs  of  ftictors 


1 

144 

2 

72 

4 

86 

.r-l-8 

X     13 

And  since  the  sign  of  the  third  term  is  — ,  we  write 
the  sign  of  the  second  term  (in  this  case  — )  abovo 
the  column  of  larger  factors,  and  the  other  sign  (of 
the  pair,  d=)  above  the  other  column. 

Ex.  27. 

1.  x^-bx-U\  x''-~S)x  +  14  ;  x!"  +  7a;  +  12. 

2.  x^-^x  +  lb;  a;2-19a;  +  84;  a;2-7.r-60. 

3.  4.1-'- 2.1- -20;  Oa;^  -  150:i-  + 600. 

4.  \x^-\-^x-?>Q>]  25:c^  +  40:p+15;  9.r«  -  27^;^  + 20. 

5.  J^a;^+1}^+12;  16a;*  -4. i'^- 20. 

6.  x'  -  {a'  -h  h')  x^  +  a^  V  ;  4  {x  -\-  yf  -  4  (a;  +  y)  -  99. 

7.  {x^^fy-{a^-h''-){x'^ii')-o}h\ 

8.  (a  +  Z>)''-2c(rt  +  i)-3c^ 

9.  (a;  +  yy  +  2(a;'  +  /)(^  +  y)  +  (^-2/T 


84 


FACTORING. 


10.  (a  +  hf  ~  4  nh  {a  -f  h)  -  (a^  -  by. 

11.  {x'  -I-  xy  4-  y'f  +  .-r'  -  y"  —  5a-y  —  2?/  -  2a;^ 

12.  a"' -'2,a{h-c)~2>{h-  c)\ 

1 3 .  (.r'  +  :'/')'  +  2  a'  {x^  +  /)  -}-«"-  h\ 

14.  (.f'-lO.if -4(a;"^-10a;)-96. 

15.  (.1'^  -  14:r  +  40)'^  -  25 {x"  -  14a:  +  40)  -  - 150. 

16.  {x'  -  xy  +  yy  +  2.ry  (x^  -  xy  +  2/'0  "  '^--^''y'- 

17.  z'-^z'-V^-  x*--2x'   -3;  9a:«  + Oa;^'    ■  lOy*. 

18.  6''^'»  +  c"* -  2  ;  .x-«-.t'-2;  .x-"*  -  2a.-'"?/** -  8y^". 

19.  x'"'~{a  —  h)x"'y''-ahy'"\ 


§  15.  Trinomials  of  the  form  ax'^-\-bx-\-c  (a  not  a  sr^nare) 
may  sometimes  be  easily  factored  from  the  following  con- 
siderations : 

The  product  of  two  binomials  consists  of 

1.  The  product  of  the  first  terms  ; 

2.  The  product  of  the  second  terms  ; 

3.  The  algebraic  sum  of  the  products  of  the  terms  taken 
diagonally. 

These  three  conditions  guide  us  in  the  converse  process 
of  resolving  a  trinomial  into  its  binomial  factors. 


Examples. 

1.    Resolve  6.r  —  13 :r?/  -|-  Qy''. 

Here  the  factors  of  the  first  term  are  x  and  6  a;,  or  2x 
and  Sx;  those  cf  the  third  term  are  y  and  6y,  or 
2y  and  3  ?/.    These  pairs  of  factors  may  be  arranged : 

(1)  (2)  (3)  (4) 

X  2x  y  2?/ 

6a;  3a;  6y  3?/ 


FACTORING. 


85 


2x\ 


- 150. 
Ixhf. 

'    loy. 


(a  not  a  square) 
following  con- 


tlie  terms  taken 

onverse  process 
jtors. 


X  and  6a;,  or  2.:c 
re  y  and  6?/,  or 
ay  be  arranged : 

;4) 

»2/ 


Now,  we  may  take  (1)  with  (3)  or  (4),  or  (2)  with  (3) 
or  (4) ;  but  none  of  these  combinations  will  satisfy 
the  third  condition.  If,  however,  in  (4)  we  inter- 
change the  coefficients  2  and  3,  then  (2)  and  (4)  give 

3y 


and 


where  we  can  combine  the  "  diagonal  "  products  to 
make  13,  and  the  factors  are 


>j  X 


3y 


an( 


3a:  —  2?/. 


Th(}  coefficients  of  (2),  instead  of  those  of  (4),  might 
have  been  interchanged,  giving  the  same  result. 

2.  (Sx'—l^xy-^Qif. 

Here,  comparing  (2)  and  (3),  Exam.  1,  we  see  that 
their  diagonal  products  may  be  combined  to  give 
15,  and  the  factors  are  2x  —  y  and  3a;  --  (Sy. 

3.  (Sx''  —  2'^xy-\-^y\ 

Here,  again  referring  to  Exam.  1,  we  see  at  once  that 
it  is  useless  to  try  hoik  (2)  and  (4),  since  the  diag- 
onal products  cannot  be  combined  in  any  way  to 
give  a  higher  result  than  13  a;?/.  But  comparing  (1) 
and  (4),  we  obtai:  ,  by  interchanging  the  coefficients 
in  4.  ^._3y 

and  6a;  — 2?/, 

which  satisfy  the  third  condition. 

Oi',  we  might  interchange  the  coefficients  of  (3),  and 
take  the  resulting  terms  with  (2),  getting 

2a;  — 6?/ 
and  3a;—    y. 


% 


I 


86 


FACTORING. 


4.  Gar'  +  SSary-Gy^ 

Here  the  large  coefficient  of  the  middle  term  show.s  at 
once  that  we  must  take  (1)  and  (3)  together.  Inter- 
changing the  coefficients  of  (1),  we  have 

Qx—    y 
and  a:  -f  G  y. 

The  same  result  Will  be  obtained  by  interchanging  the 
coefficients  of  (3). 


1.  6  a:' -  37^:7/ -f-Gyl 

2.  6ar'  +  9a:y-6yl 

3.  56a;'-7Ga;y  +  20y^ 

4.  5Ga:'-36a:y  — 20?/^ 

5.  56a:'-1121a:y  +  20yl 

6.  5Ga;'-68ary4-202/'. 

7.  56a:'-558ary-20y^ 

8.  56a;'  +  3Ga;y-20y^ 

9.  56a;'-67?7/  +  20y. 
10.    SGaj'-f  3a:y-20y^ 


£jX.  28. 

11.  G  x^  —  IG  a;y  —  G y'. 

12.  Ga,-2  +  5a:y-62/^ 

13.  5Ga;'  +  5G2^-?/  +  20?/'. 

14.  5Ga:'-122a:2/  +  20?/-. 

15.  56a;'-102a-y-20yl 

16.  56a;'''-229a:y  +  20?/l 

17.  56^^-94x7/4-20^1 

18.  5Ga;'-27Ga-y-20yl 

19.  3Ga;' -33a:y- 153/1  ^ 

20.  72a:'^-i9a:y-40?/l 


§  16.  Generally^  trinomials  of  the  form  a3?-\-hx-\-c{a  not 
a  square)  may  be  resolved  by  Formula  [A] ;  thus, 

Multiplying  by  a  we  get  o^a?  -\-  bax-\-ac.  Writing  z  for 
ax,  this  becomes  z^-\-bz-\-  ac.  Factor  this  trinomial,  restore 
the  value  of  z,  and  divide  the  result  by  a. 

Examples. 
1.    6a;'  +  5a;  — 4. 

Multiplying  by  6,  we  get(6a7)'M-5(6a:)-24  or  z'+52-24. 
Factoring,  we  get  (3  —  3)  (2 +  8);  hence,  the  required 
factors  are  ■J-(6a;-  3) (6a; -f- 8)  =  (2a;—  1) (3a: +  4). 


FACTORING. 


8; 


term  hIiowh  at 
retlicr.     Iiiter- 


2.  (j.v'-lSx//'\  (ji/\ 

Factoring  z^  —  13 zy  -f  36  ?/,  we  get  (z      4?/)  (z  —  9?/) ; 
hence,  the  required  factors  are 
i(6a;-4y)(G^-9y)-(3:r-2y)(2a;-33/). 

3.  33-14a:-40a:^ 

Factoring  1320  -  14  s  -  z^  we  get  (30  -  2)  (44  +  2): 
hence,  the  required  factors  are 
^V(30-40.rj(44  +  40a,-)-(3-4x)(ll-f-10.r). 

Note.   Tho  factors  may  conveniently  be  arranged  in  two  columns, 
each  with  its  appropriate  sign  above  it. 


Exam.  1,  above  : 


—  + 

1  24 

2  12 

\{(\x  -  3)(Ga;  +  8)  =  (2a;  -  l)(3a;  +  4). 


: 


Exam.  2,  above :  1  36 

2  18 

3  12 

l{e^x  -  4)(6a;  -  0)  =  (3.c  -  2)(2.t  -  3). 

Another  method  of  factoring  trinomials  of  the  form  ax"^  +  bx  +  c 
is  as  follows : 

Multiply  by  4  a,  thus  obtaining  ia'^x'^  +  iabx  +  4ae.  Add  6^  —  6', 
which  will  not  change  the  value,  4.a^x^  +  4a6x  +  &'  —  6^  +  4ac ;  by 
[1]  this  may  be  written  {2ax  +  bf  —  {b^  —  4ac).  Factor  this  by  [4], 
and  divide  the  result  by  4  a. 

Example.  Fa.ttor  5Gx2  +  137  a;  -  27,885,  Multiply  by  4  x  56  or 
2  X  112,  112^x2  4  2  X  137  X  112a;  -6,246,240.  Add  137^  -  137^;  then, 

1122x2  +  2  X  137  X  112x  +  1372  -  (I372  +  6,246,240) 
=  (112x  +  137)2-6,265,009 

-  [(112a;  +  137)  +  2503]  [(112a;  +  137)  -  2503] 

-  (112x  +  2640)  (1 12  X  -  2366). 

We  multiplied  by  4  X  56 ;  we  must,  therefore,  now  divide  by  that 
number.     Doing  so,  we  obtain  as  factors  (7x  f  165)  (8  x  —  169). 


88 


FACTORING. 


1.  10a;'  +  .r-21. 

2.  lOa;^ -29a:-21. 

3.  10a;'^  +  29a;-21. 

4.  Gx'~S1x-\-55. 

5.  12a' -5a -2. 

6.  12a;' -37a; +21. 

7.  12a:' +  37a; +  21. 


Ex.  29. 

8.  15a«+13a''i'-205*. 

9.  Ux'-x—l. 

10.  9x^i/-Sxi/-Cy7/. 

11.  4a;'  +  8.ry  +  3?y'. 

12.  3^>'a;'-7^>ar''-3a;\ 

13.  Qx*-xh/  —  Sbi/. 

14.  2a;*  +  a;' -45. 


15.  4a;*-37a;'y'  +  9/. 

16.  4(a;  +  2)*-37ar^(a;+2)'  +  9a;*. 

17.  6(2a;  +  3y/  +  5(6a;'+  6x1/  -  6f)  -  6(3a;  -  2i/)\ 

18.  6(2a;  +  3^')*  +  5 (6a;'  +  5a;y  -  61/')'  -  6(3a;  -  27/)*. 

19.  6(.^'+a;y+?/')'+13(a;*+a;'y'+y*)-385(a;'-a7+?,'0'- 

20.  21(a;'  +  2a;?/+2?/')'-6(a;'-2a;y  +  f':/)'-5(a;*+4y*). 

Extended  Application  of  the  Formulas. 

§  17.  The  methods  of  factoring  just  explained  may  bo 
applied  to  find  the  rational  factors,  where  such  exist,  of 
quadratic  multinomials. 

Examples. 

1.    Resolve  12a;'  -  xij  ~  20?/'  +  8a;  +  41  y  -  20. 

In  the  first  place  we  find  the  factors  of  the  first  three 

terms,  which  are 

4a;  +  5y 

and  Sx  —  Ay. 

Now,  to  find  the  remaining  tenns  of  the  required  fac- 
tors, we  must  observe  the  following  conditions : 


FACTORING. 


89 


9,aJ^h^-2W. 


iRMULAS. 


1.  Their  product  must  120. 

2.  Tho  algchraic  sum  of  the  products,  obtained  hy 
multiplying  them  diagonally  into  the  ?/'s,  must     4 1  //. 

3.  The  sum  of  the  products,  obtained  by  multiplying 
them  diagonally  into  the  a;'s,  must  =  8a:. 

We  see  at  once,  that— 4,  with  the  tirst  pair  already 
found,  and  -f '-\  with  the  second  pair,  s.itisfy  the 
required  conditions  ;   and  lience  the  factors  are 

4  a: +  5?/  — 4 

and  3a;  — 4?/ -f- •'^. 

Here  the  factors  of  jl*^  -^-p^l  "  2</'^  are 

p-]-2q 
and  P~  9.' 

Now  find  two  factors  which  will  give  —  Sr'^,  and  which, 
multiplied  diagonally  into  the  j'^'s  and  (/'s  respec- 
tively, will  give  2p?'  and  7  qr ;  these  are  found  to  be 
—  r,  taken  with  the  first  pair,  and  -}-  3r,  taken  with 
the  second  pair.     Hence,  the  required  factors  are 


and 


p-{-2:/  —  r 
J)-  q-{-Sr. 


The  work  of  seeking  for  the  factors  may  be  conveniently 
arranged  as  follows : 

-Reject : 

1.  The  terms  involving  z; 

2.  The  terms  invoking  ?/ ; 

3.  The  terms  involving  x; 
and  factor  the  expressioi  that  remains  in  each  case. 


90 


TACTORINO. 


1.  :^:»  I  xy-  Ly      (x     y/)(a;-f  2y). 

2.  x''\-2 xz  ~  3 z'      (x'  - f-  3 z)  (x-z). 

8.   -  2  v/  H  7  v/2  -  3  2»  -  -  ( ^  y  f  3 2)  (2y  -  2). 

Arruiige  tlieso  tlireo  pairs  of  factors  in  two  sets  of  ihrcc 
factors  eacli,  ])y  so  selecting  one  factor  from  eacli 
paii"  that  two  of  each  setof  tliree  may  liave  the  sjime 
coefRcient  of  2*,  two  may  liave  the  same  coefficient  of 
;//,  and  two  the  same  coefficient  of  z  {(-orffwicnl  incl ail- 
ing s'uj)i).     In  this  example  there  are 

x-y,       a: -I- 32,     -2/-I-32, 

and  X  -f-  2y,     a:  —  2,         2y  —  2. 

From  the  first  set  select  the  common  terms  (including 

signs)  and  form  therewith  a  trinomial,  x  —  y  I  82. 
Repeat  with  the  second  set,  and  we  get  x-\-  2y  —  z. 
.'.  x"  4-  .ry  -  2y^ -f  2.r2  +  7y2    •  3 2' 

-(:r--y-f32)(.H-2y-2). 

4.  3.t'^  -8.x-y      3y'^  +  30.i'  +  27. 

1.  3a;'^"8a;y-3y'^  =  (3a;  +  y)(^-3y). 

2.  3a;M-30.r  +  27  =  (3;r  +  3)(a--h9). 

3.  -3y^  +  27-(y  +  3)(-3y  +  9). 

.'.  the  factors  are  (3a:  -fy  -f-  3)  (.r  —  3y  ~f  9). 

5.  (See  ~7ah  +  2ac  -  20^^  -{-(jUc-  i3c\ 

,       1.   Qa''-7ab~20h'=(2a-~5h)C6a  +  4:b). 

2.  Ga''-{-2ac~iSc^  =  (2a  +  6c)(^a~Sc). 

3.  -20b'  +  GUc-48o^  =  {-5b-^6c){U-Sc). 
.'.  the  factors  are  (2a- 55 +  6c)(3a  +  46  -  8c?). 

To  find,  where  such  exist,  the  factors  of 

ax^  +  bx7/  -|-  cxz  +  et/^  +  gyz  +  hz^. 
Multiply  by  4a : 

4  a^  x^  -f  4  ahxy  -f  4  acxz  -j-  4  aeif  -\-  4  agyz  -|-  4  a/i2^ 


PA<^TORTNa. 


91 


Select  the  terms  contjiiiiinj^' a^-,  aiid  completo  the  squaro: 

till  18, 

4  rt' x^  -{-  i  ahxy  +  4  ncxz  +  b^  f  ■]  -  bcxz  \-  C'  2" 

=  (2fw;-f-6y-f-C2)» 

-    [(/;'      4  «t')  y'  -\-  2  (ic-  -  2 ag) yz  +  (c*  -  4  ah)  z'\. 

If  the  part  within  the  double  bracket  is  a .H(juare,  Hay 
{nwj -\-i\zY,  the  given  expression  can  be  written 

{;inx -^y  by  ■\- czf    -{my  +  nz)\ 

wliicli  can  be  i\xctored  by  [4].  Factor  and  divide  the  result 
by  4  a.  If  the  part  within  the  double  bracket  is  not  a 
s(|uare,  the  given  expression  cannot  be  factored.  If  h  and  c 
are  1>otk  even,  multiply  by  a  instead  of  by  Aa,  and  the  scjuare 
can  be  completed  without  introducing  fractions.  If  c  be 
less  than  a,  it  will  be  easier  to  multiply  by  4c  instead  of 
by  4«,  and  select  the  terms  containing  y.  A  similar  remark 
applies  to  h. 

This    method    can   evidently  be  extended   to   quadratic 
multinomials  of  any  number  of  terms. 


Examples. 

1 .    Resolve  x^  -{-  xy  \-  2xz  —  2 y"^  -{-  7 yz  ~  S  z^  into  factors. 
Multiply  by  4  : 

4a,-''  +  4:xy  +  80:2  -  8/  -f  2Syz  -  122^ 
Complete  the  square,  selecting  terms  in  x : 

4.i''-f-4a:?/-h8:r2  +  yH  4v/2  +  42'^ -9v/''-|-24?/2-162'' 

-(22-  +  2/  +  22)^    -(3y--42)'' 

-  [(2  .T+  y  -f  22)  +  (3y  -  42)]  [(2:.' + y+  22)-  (3y-42)] 

=  (2:c-h4y-22)(2a:-2y+62)  --^  ^x-{-2y-z)(x   y+Sz). 
.•.  the  factors  are  (x  -{-  2y    -  z)(x  —  y  -{-3z). 


92 


FATTORTNO. 


2.  Ga"      'lah  \  2ac  -  20 i" -f  04  Ac  -  486-^ 

Multiply  by  4  X  G  =  24: 

144a'^  -  IGSai  +  48rw  -  480i-  -}-  153Gi6>  -    1152<? 
=  (12a  -  76  -f  26'/  -    520 6'^  f  15G46c-  llOGc'^ 
-  (12  a  -  7  ^*  -f  2  cy  -  (23  b  -  34  rf 
^  (12a  +  lG/>  -  32d')(12a  -  30i  +  3Gf) 
=  24(3a  f  4Z»  -  8c')(2a  -  bb  +  Gc). 
.•.  the  fjictorH  are  3a -f  46  —  8f  and  2a      TiA   [   G<'. 

3.  x'  \-l2xy-\-2x2-\~2(j7/-H7/z~  dz" 

--  (.f'^-f  12^y4-2a--f-3Gy'-j- 12^2+2*0    10/-20y2  -lOs'^ 
-(.-r  +  6y  +  2)^-[(y  +  z)Vl0r 

=  [^  +  (G+vio)y4-(Vio  I  i>] 

x[^'+(G-viO)y--(Vlo  -1)4 

4.  3a^  I  10a6    -14ac4-12a(^-86*^-86r/-^  86'^      8fY7. 

Multiply  l)y  )),  not  4x3,  Hince  the  coeflicients  of  the 
other  terma  in  a  are  all  even  : 

Oa'^  -f  30a6  -  42ae  +  SGad  -  24h''  -  246^/+ 246''^-  24cd. 
Select  the  terms  containing  a,  and  com])lete  the  scpiare  : 

(3a4- 56 -7^  +  6^0' 

-  49  b'  +  70  be  -  84  hd  -  -  25  r'   f-  GO  cd  -  -  3G  ^/"^ 

=  (3 a  +  5  6  -  7  c  +  G  t/)*^  -  (7  6   5  6'  +  G  f/)'^ 

=  (3a  +  126  -  12c  +  12  J) (3a  -  26  -  2c) 

=  3(:a  +  46-4cH-4ri)(3a-26  -2e). 

.•.  the  factors  are  a  +  46  —  4c-|-4c/  and  3 a  —  26  —  2c. 


Ex.  30. 

1.  7.'r'-a:y~G7/^-G^ -20y-lG. 

2.  20a:»-15a:?/-5y^  — G8:r-42y-88. 

3.  3a;*  +  a:^y'-42/*  +  10a;^-172/^-13. 

4.  20a;'^-20y'  +  9a:y+28:r  L35?/. 


FArTORINO. 


on 


6.  72^-^     8y'-|  55.ry  f  12y-ir)9.r-h20. 

6.  t'  --  xy  —  12?/  —  5  .r  —  15^. 

7.  8.7.-'  f  18r//  -h  9?/  -I-  Ixz  -  2». 

8.  (') x'  \-  G?/  -  13.77/  -  82"'      2?/2  +  8a-2. 

9.  Or*    lO/Z+lLt'/-  252M-10yH25yV    152-M  lO.r':'. 

10.  15 .7*      1 0 y      22 x^ >f  +  1 5 2*  i   1 4 // c'  'V  50 .r' s^ 

11.  4a»-15//-4«/>  -2l6''--3G^'6'      8fir. 

12.  a*  +  ^''*  +  ^'*-2a''Z;'-2Z'"'c''-26'^«^ 

§  18.   Trinomials  of  the  form  ax^  4  ^>'-^"'  +  ^  can  always  be 
broken  up  into  real  factors. 

If  a  and  c  liave  difTcrent  signs,  the  exprossion  may  be 
factored  bv  §  10. 

If  a  and  o  are  of  the  same  sign,  three  cases  have  to  be 
consider«»d  : 

(i.)    />-2VM. 

(ii.)    /v>2VM- 

(iii.)   /><2VM- 

Case  I.     h  -  2  V(«c).     This  ca.se  falls  under  §  12,  For- 
mula [1],  where  examples  will  be  found. 

Case  II.    Z>  >  2  ^{ac).     This  case  falls  under  §  10,  where 
examples  will  be  found. 

The  following  additional  examples  ars  resolved  by  the 
second  method  of  that  section  : 


I 


Examples. 
1.    4a;*  +  5:r'?/2+2/*. 

Here  we  see  that  (J/)^  will  make,  with  the  first  two 
terms,  a  perfect  square,  and  we  therefore  add  to  the 
given  expression,  {\lfY~^\y^f- 


94 


FACTORING. 


The  expression  then  becomes 

2.  3a7*  +  6a."^  +  2. 

Here,  multiplying  by  4  X  3,  and  completing  the  square 
as  in  Exam.  1,  we  have 
36a:*+72a;2  +  6^-}-24  -G'^ 
=  (6:i-^  +  G)'^-  12 

which,  divided  by  4  X  3,  gives  the  required  factors. 

3.  ax*  -\-  hx^  +  c. 

Proceeding  as  in  Exam.  2,  we  have,  by  multiplying 
by  4a, 
ax*^  +  bx"^  \-  c 

=  i^cex"  -^^ahx'^-h^-V^^iac)  h-  \a 
--=  [2ax''-\-h-\-^{I?-Aac)']  [2ax'-\-h-^{h'^Aac)]  ^4a. 


Ex.  31. 

1.  :u*  +  7a;'  +  l;  4a;*  +  14a;2  +  l. 

2.  x'-\-1x'''if-\-i/]  Zx''-\-bx'if-\-y\ 

3.  4.r^  +  10a;M-3;  ?,  {x -\- y)' -^  b  z\x  ^  yf -]- z\ 

5.  4.r*  +  9:r^?/M-i|2/*;  4(rt+5)*  +  106'^(a-f^)'^  +  3f^ 

6.  3a;*  +  8a;^yM-43Ly;  30.1-^  96.r^ +  55. 

7.  5a:*  +  20a;^  +  2;  4a*+12a'  +  l. 

8.  4(a;  +  ?/)*-|-12(a;H-yy2'4-2*;  5a;*  +  20./;'^y'H  2^. 


FACTORING. 


05 


eting  the  square 


equired  factors. 


by  multiplying 


10.  2a:*+12:r'+15;   1x* -\-^''x'-\-45. 

11.  8x'-\-S(jx'f  +  29y';  7 x* -{-20x''7/ -20i/. 

12.  7{a-by  +  l(j(a-hyc'-\-5c*;  ^a* -i-Sa'b'' i-b\ 

13.  8.rM-6.r'y"'  +  2y*;  d{a-i- by -\- C,(^c'- by +  2(a-  by. 

14.  i9a*-S^a'b'-{-22b*-  25 m* -{•  60 m' ?i' '\- 27 7i\ 

15.  49(m  +  ??y-84(m^  -7i'0^  +  22(w-/?)\ 

Case  III.  b  <  2'^{an).  This  case  may  be  brought 
under  §  13. 

The  following  examples  illustrate  the  process  of  reduc- 
tion and  resolution  : 

1.  0^-1x^  +  1.  Examples. 

We  have  to  throw  this  into  the  form  a^  —  b'^: 
X*-  7x'  +  l  =  (x'i-iy-9x' 
=  (x''-}-l  +  Sx)(x'+l-Sx). 

2.  9x*  +  3x''f-\-^y*  =  {Sx''-{-2fy~9x'y' 

=  (3a:' 4-  2i/  -  3xy){3x'  +  2fi-3x7j). 

3.  x'-\-7/  =  {x''-^y'y-~2x'y' 

-  (x'  +  f  +  xy  V2)  (^'  +  y^  -  ^y  V2). 

4.  a;*-Ja;'?/'  +  3/*  =  (a;'  +  y7-fa;'7'' 

=  (^'  +  2/' + f  ^y)  (^'  +  y'  ~  f  ^y). 

5.  aa;*  +  ^-^-'  +  <?  =  (^V«+V^)'-  (2v^ 


^>)a;2 


=-  \x'^a-]-^c-(\2-^ac~~b)x\ 


§  19.  It  is  seen  from  these  examples  that  we  have  merely 
to  add  to  the  given  expression  what  will  make  with  the 
first  and  last  terms  (arranged  as  in  Exam.  5)  a  perfect 
square,  and  to  subtract  the  same  quantity.     In  Exam.  2, 


i 


i. 


96 


FACTORING. 


for  instance,  the  square  root  of  9x*  =  Sx'',  the  square  root 
of  4y*=2?/'';  hence,  Sx'^-\-2i/^  is  the  binomial  whose  square 
is  required  ;  we  need,  therefore,  12a;'^3/'^;  but  the  expression 
contains  ^tx^i/]  hence,  we  have  to  add  and  suhtrcct  \1x^y^ 

Hence,  we  derive  a  practical  rule  for  factoring  such 
expressions : 

1.  Take  the  square  roots  of  the  two  extreme  terms,  and 
connect  them  by  the  proper  sign ;  this  gives  the  first  two 
terms  of  the  required  factors. 

2.  Subtract  the  middle  term  of  the  given  expression  from 
twice  the  product  of  these  two  roots,  and  the  square  roots 
of  the  difference  will  be  the  third  terms  of  the  required 
factors. 

6.  a;*  +  ^a:V  +  3/*. 

Here  y/x^  =  x^,  -y/y*  =  y^,  and  the  first  two  terms  of 
the  required  factors  are  x^  -{-i/^ ;  twice  the  product 
of  these  is  -|-2a;^y^,  from  which,  subtracting  the  mid- 
dle term,  ■^x^y'^,  we  get-^.t'^y^;  the  square  roots 
of  this  are  ±  ^xy.     Hence,  the  factors  are 

x^-^-y^dz^xy. 

Note  that,  since  -y/y*  =+3/^,  or  —y"^,  it  may  sometimes 
happen  that  while  the  former  sign  will  give  irrational 
factors,  the  latter  will  give  rational  factors,  and  con- 
versely. 

7.  x*~llx^y^  -{-y*. 

Here,  taking  -^-y^,  we  have 

^'^  +  y'^  +  '^y  ^/^^  a^id  rt-'  -{-  y'^  —  xy  -y/lS. 
But,  taking  ~y^,  we  have 

x^  —  2/^  H-  3  xy  and  x"^  —  y"^  —  S  xy. 

Sometimes  both  signs  will  give  rational  factors. 


FACTORING. 


97 


8.    16x*-17x'i/-\-y*. 

Here  we  have  (4a;'  +  y'  +  3 X7j)  (4.^^  +  /  —  3 xy), 
and  also        (4  a;''  —  ^  +  5  xij)  (4  x^  —  y'^      5  a;y) . 


me  terms,  and 
IS  the  first  two 

xpression  from 
le  square  roots 
f  the  required 


two  terms  of 
;e  the  product 
3ting  the  mid- 
square  roots 
are 

|ay  sometimes 
jive  irrational 
;ors,  and  con- 


Ex.  32. 

1.  x*  +  2x'i/-{-0i/;  x'^   x'y'  +  i/;  x^-^-x'if  +  T/* 

2.  x^  +  ii/;  Ux*  +  y'-x'7/-  ^x'-^-y*. 


X" 


X 


-7x'  +  l;  x*  +  d;  \x*  +  y*-Sx'y' 


5.  9/-x*+nxhf;  a;'  +  4/;  x*i-4:X^-{-lQ. 

6.  4a;*4-y*-8ia:Y;  x* +  y* --^x'l/ ]  4a:* +  1. 

7.  a;*"'+64y";  a;*'"4-4y"*;  i-^H  A^-^Ja;'?/'. 

8.  4a:*-8a;''  +  l;  7a;y  -  Ja;*- 36y*;  a;*  +  a*/. 


m 


2^4 


b;— 5  .,4ni 


a;*  +  w'  y*  -  (2  mn  +  p)  a;''  t/'^  ;  a:***  +  2'«---'  y 


10.  16a;*-25a;^H-9;  4a;*  -  16a;'  +  4;  13a;y- 9a;*-4y 

11.  4a;*-12f|a;'y'  +  9y*;  a;*  +  6a;2  +  25. 

12.  a*  +  6*  +  (a  +  ^)*;  1  + a* +  (!  +  «)*. 

13.  (x  +  yy-7z'(x-{-yy~^z\ 

14.  (a  +  Z.)*  +  7c'  (a  +  i)'  +  c\ 

15.  16a*  +  4(6-c)*-9ft'(6-c)». 

16.  4:(a  +  by  +  d(a-by -21(0"- by. 

17.  (a;'  +  y'-a;y)*-7(a;'  +  y^y  +  (^  +  yy. 

18.  (a' +  a6  4- i')*  +  7  (a»  -  67^  +  (a  -  Z»)*. 

19.  16a*  +  4a'  +  l;  a;*-41a;2^^ig 

20.  (a'  +  1)*  4-  4 (a'  -f  1)' a'  -f  16 a* ; 

(a;  +  1)*  +  2  (a;' -  1)' 4- 9  (^  -  1)*. 


98 


FACTORING. 


§20.   We  can  apply  [4],  §  13,  to  factor  expressions  of  the 
form  ex*  -j-  hx^  -{-  rbx  —  r'^a.     This  may  be  written, 

a  (x'      r')  -I-  bx  {x'  [-  r)  -  [a  {x"  -  r)  -\-  bx]  (x"  +  r). 


Examples. 


1.    G.r*4-4r'+12 


X 


54 


^e>{x'-^)-\-^x{x'-\-?>) 

-(:i-^-f3)[6(a;^~^3)  +  4^] 
=  (a;^  +  3)(6a;'  +  4a;-18). 

2.  \lx'+l0a^-^0x-l1(S 

=:ll(:^;*-16)  +  10.^•(.^2-4) 
'  -(:r^-4)[ll(2'H4)  +  10:f] 
=  Ix"  -  4)  (11  x'  +  10:r  +  44). 

3.  40a:*  +  30a;'  +  C0a;-lG0 

=  10(4a;*  -  16)  +  lbx(2x'  +  4) 
=  (2a;^  +  4) [lOi^x"  -  4)  +  15x] 
=  (2a;^  +  4)(20a;^  +  15a;  -  40). 

Note.   To  determine  r,  take  the  ratio  of  the  coefficient  of  a;'  to  the 


coefficient  of  x. 

Resolve  into  factors : 

1.  x'-\-2x^-^^x-'d. 

2.  2x'-{-2x^-\-^x-l^. 

3.  a;*  +  3a;'+12a;-16. 

4.  3a;*  H-ar'- 4a; -48. 

5.  5a;H4a,-'-12a;-45. 


Ex.  33. 


6.  10a;*  +  5a;'-f30a;-360. 

7.  i.r*  +  20ar"'  +  4a;-y^. 

8.  25a;*-40rH8a;-l. 

9.  37ia;*-30a;='+48.r-96. 
10.  G3a;*-39a;^+52a;-112. 


11.  810a;*  +  -8^a;'  +  |a;-2}. 

12.  242a;*-33a;^-3a;-2. 


FACTORING. 


90 


cient  of  x^  to  the 


13.  \x'  +  j\r^-^\r      ,V 

14.  80a;*      32rV-{-04.ry-320/. 

15.  2ix*  -  I2x'f/  +  30.r/  -  150?/*. 

16.  2.7-*  f  ^x^(/  -  8.?^    -  512?/. 

17.  ll.r*+10.f^-    12:i-15|i. 

18.  40.r*  f  30.ir'-|  G0:r~160. 

19.  13.t*  -  12x'i/-\-  Tlxif  -  4G8/. 

20.  3.-l;*-f-3.t-''y+12.^'?/*-48?/. 

21.  5.i-*4-4:r'?/  — 12.r/  — 45?/. 

22.  4.7;*-14r'?/  +  28a;?/'-10?/. 

2.3.    .x-*-f  80:r■\?/+16.^y-^V?/• 
24.    2a;*-:r'?/  +  62y-72?/*. 

§  21.    Formulas  [1]  and  [4]  may  sometimes  be  applied  to 
factor  expressions  of  the  form 

ax^  +  hx"^  -f  cx"^  -{-  rhx  -\-r'^a. 
This  may  be  put  under  the  form 

a  (:r*  +  r")  -f-  hx  {x""  -f  r)  +  ex" 

=  a  (x^  +  rf  +  ^^'  (a^"'  +  r)  +  (c  -  2  ar)  x\ 
which  can  sometimes  be  factored. 


'+52:^-112. 


Examples. 


x'  -f  6.r*  +  21  x"  +  162a;  +  729. 

3  +  6a;(a;^  +  27)+27a;' 
27)'  +  ijx{x''  -I-  27)  -h  Oo.-'  -  ZCix' 
27  +  3a;)^  —  36a;^  which  gives  the  factors 


(a;'  +  27  +  3a;)^~36a;^  whic: 
-3a;  +  27anda;^-f  9a;  +  27. 


100 


FACTORING. 


2.    a,*-f-4a;='4  4:cM-20a;-f  25 

■~~-  (.r^  +  5)^  -f  4  X  (.c^  -I  -  5)  -  G  x" 

^.  {x""  }-  5)^^  +  ^x{x''  +  5)  +  ^x'  ~  lOx"" 

=  lx'-\-b-\-'2.x-  x^\0)  (a;'  +  5  +  2  x  +  .r  VlO). 


Ex.  34. 


Resolve  into  factors 


1.  .T*  -  Gr"'  +  27a.-'  -  lG2x'  +  720. 

2.  .r*  +  2a;'  +  3.r'  +  8a;+lG. 


.r 


x^ 


+  a;'  +  a;'  +  a;  +  l. 
—  4a;'^  +  a;''  — 4.r+  1. 


5.  4a;*-12ar'-Ga;'^-  12;r  +  4. 

6.  .r*+14.r"'-25a;'-70a;+25. 

7.  16a:*- 24r^-lGa;H12a;  +  4. 

8.  .x'*  +  5ar'-lGru'  +  20a;  +  16. 


X' 


+  6a:'-lla;'^-12a:  +  4. 


10.    x^-\-^a^y-^x'if-\-l2xif-\-^i/ 


11. 


X 


+  6ar'-9a:'^  — 6a:  +  l. 


12.  .r*  +  4^;^?/  -    19a:'?/'  +  4ay  +  y\ 

13.  4a:*  +  4.rV--  G5a:'2/'-  10a:y''  +  25?/*. 

14.  a:*  +  Go:"'?/ —  9a:'?/'  — Go:?/^  +  y*. 

15.  a:*  +  6a:'y  +  10a;'y'  +  12a:?/'  +  4y*. 

16.  9a:*  +  18a:='y-52a:'?/'-12a;?/^  +  4y*. 

17.  11  a:*  +  10.r^?/  +  39^  :i-'?/'  +  20a:/  +  44  y* 


Factoring  by  Parts. 

§  22.   To  factor  an  expression  which  can  be  reduced  to 
the  form  a  X  F{x)-\-h  Xf(x). 


FACTOR!  N'r;. 


101 


e  reduced  to 


WIk.'u  the  expression  is  tlius  arranged,  any  factor  com- 
mon to  a  and  />,  or  to  F{x)  and/(.f),  will  be  a  factor  of  the 
wlioh^  expression.  The  method  about  to  be  ilhistrated  will 
1)0  found  useful  in  cases  where  only  one  power  of  some  let- 
tt"-  is  found. 

Examples. 

1.  V'A.Qiov  acx^ —  ahx -hc-X'\  h'c. 

Here  we  see  that  only  one  power  of  a  occurs,  and  we 
therefore  group  together  the  terms  involving  this 
letter,  and  tliose  not  involving  it,  getting 

a  {cx"^ '  -  hx)    -  hc'^x  +  h'^c 

=  ax  (ex  —  b)  —  he  (ex  —  b)  ~.  (ax  —  be)  (ex  -    b). 

2.  Factor  771^ x^ —  mna^x  — 7nnx-\- 71^ a^. 

Here  we  observe  that  a  occurs  in  only  one  power  (a^). 
Therefore,  we  have 

—  a^  (mux  —  71^)  +  ■Wi'^o:'^  —  '77inx 
=  —  nd^()7ix  —  n)  -f-  7nx (vix  —  n) 
=  (^77ix  —  7i)  (771X  —  nd^). 

3.  Factor  2x^  +  4:ax'i- Sbx-\- Gab. 

Here  we  observe  that  the  expression  contains  only  one 
power  of  both  a  and  b.  We  may,  therefore,  collect 
the  coefficients  in  either  of  the  following  ways : 

a(4:x  i-  (jb)  +  (2x'  -\-  Sbx), 
or  i  (3 a;  +  6 a)  +  (2x''  ~\-  4 ax). 

Now  the  expressions  in  the  brackets  ought  to  have  a 
common  factor ;  and  we  see  that  this  is  the  case. 
Hence, 

a(4:X  +  Qb)  +  (2x''-^Sbx) 
=^2a(2x+Sb)i-x(2x  +  Sb) 
=  (2x  +  Sb)(xi'2a). 


■ 


102 


FACTORINd. 


:i 


4.  ahxy  -f-  b^'if  -\-  acx  ~  & 

=  a  {hxy  +  Gx)  +  '>'y^   -  c' 

=  ax  {hy  +  c)  +  {hy  -f  c)  {by      r) 

^=(by+c){ax-{-by-~c). 

5.  y"    -  (2a  -h  b)  ?/  f  (2a/»  4-  (t")  y  -  r/,V>> 

=  -  ^» (/  -  2«?/  +  a'^)  +  ?/      2a/  ]  a,'^// 
=  -  b  (7/  -  2  ay  +  a'O  f  V  (y'   -  2  ay  h  >) 
^  (2/ "  ^^)  (y  -  ^0'- 

6.  2x^y  4-  2 &.2;*  -  /ja;''y  f  4 ai^c^ y  —  a;''?/*'^  +  4  aa:v/'' 

-~2a/>»a:y''  — 2ay^ 
=  b{2x^  —  r^y  +  ^ax?y  —  2axy'^)  -f-  2a;''y  —  .r'y'^ 

4-4aa:2/'^  — 2  a?/' 
=  /^.r  (2  x^  —  x^y  -\-  ^  axy  —  2  ay') 

+  ?/  (2  or*  —  a;''^  y  +  4  aa:y  —  2  ay'^) 
=  (y  +  ^^')  (2a;^  —  a;*y  +  4  axy  —  2  ay'^). 

And  2^*  —  a:^y  -f  4 o,xy  —  2 ay' 
—  a(^xy  —  2y')  +  2a;'  —  x^y 
=•  2  ay  (2  a;  —  y)  4- a;' (2  a;  —  y) 
=  (2ay  +  a;')(2a:-y). 

7.  a;''4-(2a-^»)a;'-(2a5-a')a;-a'6 

=  6 (—  a:'  —  2 aa;  —  a^)  +  a;'  +  2aa;'  -f-  a^x 
=  -bix  +  ay  -{-  x{x  +  aY 
^{x-b){x-\-a)\ 

8.  px^  —  {p  —  q)^-\-{p  —  g)x-\-  q 

=  q{^  —  X  -{-  V) -{- pot?  —  px^  -\- px 
=  q{x^  —  X  -\-V)  -\-  px{x^  ~  x  -\-\) 
=  {px-{-q){x''~x-\-l). 


Ex.  35. 

1.  xhj  —  3? z  —  y"^  -\- yz.  3.   x^z^-\rax^ 

2.  abxy  +  i'^y*  +  aca;  —  c'.        4.   2 a;'  —  aa;  —  4  6a;  4-  2  aJ. 


d^^  —  c^. 


FACTORINC. 


lor, 


m 

1 

5. 

1 

6. 

1 

7. 

1 

11. 

(i^  If 

iMk 

12. 
13. 

4  axxf 

1 

14. 
15. 

x^y  —  x^\f 

1 

16. 
17. 
18. 

). 

19. 
20. 
21. 
22. 
23. 

}x 

24. 
25. 

26. 

27. 

28. 
29. 

-o?z^  —  d?. 

1 

4^  r% 

46a;  +  2a6. 


./•'  f-  "Ihx  +  3rt.?;  -h  ^\ah.        8.    8.r^-f42a.r  f-10/>.?;-f  ir)«/>. 


./r' 


.r* 


'.2  ..!« 


3  ...3 


a"  X 


\-d'U 


9.    ie-\-{iw--l?)x^-\hcx^ 


ax 


bKx"  +  a"*  h\      10.    f<"^  +  (ac  --  h')  x'  -  ic.r\ 
nhx'  +  (etc  -  //^O'^"'  -  (^'/  !-  ^^^O-'^'  +  #• 

.r"*  f  (a -f- 1)--^-' +  («  +  1)^' +  a. 

7Hj).ii^  '\-  ())iq  —  7)p)x^  —  [vir  -f  7iq)x  -f  nr. 

./  -  (a  -f-  h  -f  c):f'^  +  (rt^  +  ^(^  +  ac)a--  -  -  abc. 

x^  -[-(a  —  b~  c)  x^  —  (rti  —  Z>c  +  ^'a)  -"^  -  V  ^'bc. 

x^ '[-  (a-^-b  —  c)  x^  —  {be  ~  ca  —  ab)  x   -  abc. 

a^x^  —  cc'x^y  —  d^xij  +  a^y"^—  aoc^yz  +  x^z  —  xyz  +  ay'^z. 

a'-bx'  -f-  ab^xy  -f-  acdxy  +  occi??/'''  —  acfxz  —  bcfyz. 

d^x^  —  a(b  —  c)x^  i-c(a~  b)x-{-  &. 

mx^  —  nx^  y  +  rx^  z  —  mxy^  -|-  nif  —  rifz. 

avix^  +  {^nby  —  yiay  +  wicz)  a;  —  nby"^  —  7«"y/2;. 

(rt??i  —  bnn)  a?  -\-  (am  —  ben)  x-\-an-\-  nax. 


a 


b^  &  —  b"^  c^  xy  —  a^e^yz-\-  o"^  xy"^  z~  a^b"^  zx  -\-  b'^  x^  yz 


.2o,2„2 


-\~a  z  xy  —  x^y  z 


x^  —  m^  X*  —  (n  —  n^)  x^ + (f^^"^  ^  —  m"^  f^)  x^—a  {x^  -\-n^—  n) . 

I -(a-  l)x  -  (a  -  6  +  l)x''  +  (a  +  6  -  c) 
—  (b -\- c)  X*  -{-  cxr*. 


X 


a' 


3^^ 


a^  —  a^  (b—e-\-d)xhj  —  (abc  —  abd-\-avd)  xy^-\-  bcdif. 


m 


npx^  —  (n^p  —  m 


''n^ 


111 


'pq) 


X' 


(n^  -f  '>^pq  —  1^  i^q)  X  —  n^  q. 


30 .    m^  p^  x^  -f  7n^p^  x^  —  {  p'^  r^  —  (f  m^)  x^  if 

—  (^'^71^  —  q^7n^)ji^7f  —  (^n^q^  +  7i^q^x)y^ 


Kit 


FACTORIN(;. 


§23.  SoiiK'LixneH  an  expression  whieli  does  not  conif 
directly  under  the  preceding  form  may  be  resolved  by  first 
finding  the  factors  of  its  parts 

Examples. 

1 .  nhx^  -j-  (ihif  -  a^  xy  —  I?  xy. 

Here,  taking  ax  out  of  the  first  and  third  terms,  and 
by  out  of  the  second  and  fourth  terms,  we  liave 

ax{hx  —  ay)  —  hy{hx  —  ay), 
and  hence  {ax  —  hy)  (bx  —  (///). 

2.  .f*  -  {a  +  h)r^  +  (ci^b-\-  ab'')x  -  d'b\ 

Here,  taking  the  first  and  hist  terms  together,  and  tlie 
two  middle  terms  together,  we  have 

(a.-'  '\-ab)  {x'  —  ab)  -    {a-\-b)  x"  +  ab  {a  +  b)  x 
=  (x"^  4-  ab)  (x'^  —  ab)  —  (a  +  b)  x  (.tr^  -  ab) 
—-  (x^  —  ab)  [x^  -\- ab  ~  (a -\- b)  x] 
=  (:c^  —  ab)  (x  —  a)  (x  —  ^>'). 


3.    0,-'"'*  — 4a;"*  +  3 

=  .r'"'  —  a;"*  —  3  (x*^  —  1) 
=  x'^ix'"^  —  1)  —  3  (x'"  —  1) 
=  x""  (x"^  -i-l)(x'^~l)  —  S  (x" 
=  (x^^  -  1)  [x'''  (x^^  +  1)  --  3]. 


1) 


Ex.  36. 


1 .  tt^  —  ab  -f  ax  —  bx. 

2.  abx^-^-V^xy—a^xy—abif. 

3.  a;*  +  adi?  —  a^x  —  a*. 


7 .  a^—b'^-\-ax—  ac — bx-{-  be. 

8.  a^-^(l^a)ab-^b\ 

9.  a;*  +  2  xy  (x^  —  y'^)  —  y^. 


4.  a=*a:  +  2a'x2f2aa:'+A'*-     10.    a^  —  f -{- x^ -{- xy  ^  y\ 

5.  aea;H  (<^«f^— ^^)^-^^-    H-    2b  +  (b^ -- 4:) x  -  2bx\ 

6.  25a;*-5a;^  +  ar'-l.         12.   a;3  +  3a;''-4. 


FACTORINQ. 


105 


3gether,  and  the 


13.    p''-J)^q-2pr/\-2q\       20.    a^  -  4 r//>»  +  .S // 


21.  «■'"•  — Sa^f/'-f  26''". 

22.  aa^  -  (a' + /^)  x' '\- h\ 

23.  ^i>a:"'-Ga''x''-da'. 

24.  d'h^i^ahtr^  -a'c'-h^c'. 

25.  nf??i'  —  ah^  +  ^'^^^  —  '^^'^• 


14.  a-^-f-a»-2. 

15.  3aV>*      2r///-      1. 

16.  ?/      3  7/  f  2. 

17.  2a'-a'h~ah''-\'2h\ 

18.  //■"»  + />'^'«  -  2. 

19.  ;/" -^//'•s''  -2?/V"}  r'".    26.    J  — 0^2  {-27  r^*. 

27.  (.r    -  9/y  •  h  (1  -     .''■  -I- .'/)  C'-  -    //)  z  -  2". 

28 .  24  7)1^  --  28  ??i''  n  +  6  ^nn"  -  7  7i'\ 

29 .  .r"'+"  +  a-"  ?/"  +  ^•*"  y*"  +  y""^'*- 

30 .  .r*  - 1-  2  .r' y  -  «'  x^  -f  0,-'  y'  -  2  a.iy  -  y\ 

Application  of  the  Theory  of  Divisors. 

§  24.    By  Tlit'orem  I.  we  prove  that 

a;"    -  a**  is  divisible  by  x~a  ahvaya, 

x^  —  a**  is  divisible  by  x  -f  a  wlieii  n  is  evcn^ 

a?"  +  «"  is  divisible  by  x-\-a  when  ?i  is  odd. 

By  actual  division  we  find  in  the  above  cases : 


a;** 

— 

a" 

a; 

— 

a 

a:" 

— 

a" 

X 

« 

a:" 

a" 

:=  a;"-^ 


a- 


,n-2 


«  + -{-xcC 


n-2 


a 


n-l 


X'\-a 


X 


,n-l 


a;' 


,n-2 


a  + —  a:a"~'  +  «' 


,n-l 


Examples. 


1.    Resolve  into  factors  a:^  —  3/^. 

Here  a:  —  ?/  is  one  factor,  and  by  (1)  the  other  is 


(1) 

(2) 
(3) 


106 


FACTORING. 


2.  Uosolvoa'f (/>  — r)\ 

Hero  a  +  {h  —  c)  is  one  factor,  iind  by  {'X)  the  utlior  is 
a»-a(i-r)  +  (^^-^)'. 

3.  IloHolvna.''*-!  1024?/". 

This  o(|iialH  (.f'Tl  [(2y)']\  ono    factor   of  wliidi    is 
ar''4-(2yy,  and  l»y  (3)  tlic  otlior  factor  is 
(..•■')♦    -  (.r')\4y0  -j  (./•=')'^(4//7^  -  .rX-ly'O'  -I-  (4v/j' 
=  a;"  ~  4:f»/  I-  ICf^y*      G4x'^v/" -^  25G/. 

4.  Resolve  (.r  —  2?/)^  -f  (2.r  —  yY  into  factors. 
Here,  by  (3),  wo  liave 

{x~1yY  \{2x-yY 
.r  -  2y  -\'2x-~y 

-  (.1-  -  2y)^  -  (a;  -  2y)  (2a;  -  y)  +  (2;r  -  y^. 

.-.  the  factors  aro  3(.f  -    y){lx' -  ISxy  +  7?/"'). 

5.  Resolve  x^  -\-  x*y  -\-  x^y^  -f  x^if  ~\-  xy^  -f-  ?/*. 

By  (1)  we  see  that  this  equals 
a^  -'-  if  _  {di?  -\-  if)  {x^  —  if) 
x—y  ^~y 

6 .  Resolve  a;"  —  x^^  a  -\-  x^  c^  —  x^  a^  +  ^'  «*  —  oc^  (^  +  ^  «" 

—  x*"  a} -\- a?  c^  —  x^  (^ -\- xo}^  —  o}'^ . 

This  equals 

x-\r^  x-{-a 

_  (a:°  +  ft')  (^'  -  ^')  (^  -f-  ft') 
x-\-a 

-={:^^ra%x'-x^a^-^a^){x~-a){x'^xa\-a%^-xa-^a^). 


FACTOniNO. 


107 


(3)  Um^  otluT  is 


Ex.  37. 


Fatrtor  tho  following 


ir 


3    .,•« 


r"      1;  .r'  I  H;  >^(t^      127.r"';  8  |  a\t 


a 


'"•  Tm' 


04 


<r 


t'i 


.10 


0.,!l 


y 


3.    Find  11  factor  wliidi,  iiiultipli(Ml  l)y 


(C 


-f  «V>  -}-  (e'^i"^  -f  ah'  \  l>\  will  ^nyo  a*^  -  />' 


4.  By  whiit  factor  iiiUHt  .r'      4a:'v/4-  l(Jry'     04/  be  mul- 

tiplied to  give  X*  —  iZoO/  ? 

5.  Factor  x'  +  .'r'y  -|-  .r\//^  +  x*f  +  r^*  -{- a^V  j  V+  y' 


Find  tho  factors  of  the  following 


6.    (Sy"  -  2tJ  -  (.3.r*^      2?/)' ;  <'"  - 


IGh* 


8.    Z'(. 


tt 


-f  ao:  (i 


« 


')-h«^(ar-a). 


1 0 .  .r"   -  2/®  +  2  a:y  (2-*  -f-  a;"''  ^  +  y*) . 

11.  (a'-Z»c)'  +  8iV;  x*^-a*\ 

12.  .'6-'    -3a:f'*  +  3«''a:-a='  +  i'. 

13.  (r''  -f  By^'X^  +  y)  —  Ga:y(x'^  -  2xy  -f  4^^^). 

14.  Bar*  — 6a:y (2a; +  3y) +  272/'. 

15.  l~2x  +  ^x''-Sx^. 

16.  «'  4-  a^^'c  +  a^b'c'  +  a^h^^  +  ai*c*  +  ^»*c^ 


§25.  The  principles  illustrated  in  Chap.  II.  may  be  ap- 
plied to  factor  various  algebraic  expressions,  as  in  the 
following  cases : 


jT 


108 


FACTORING. 


Examples. 

1.  Find  tho  factors  of 

(a  H-  h  +  r)  {ah  +  he  +  ca)  ~(a-\-h)  (h  -f  r)  (e  -|-  a). 

1.  Observe  that  the  expression  is  syiyimetrical  with  respect 
to  a,  5,  <?. 

2.  If  there  be  any  monomial  factor,  a  must  bo  one.  Put- 
ting « =  0,  the  expression  vanishes ;  hence,  a  is  a 
factor,  and,  by  symmetry,  h  and  c  are  also  factors. 

Therefore,  ahc  is  a  factor. 

3.  There  can  be  no  other  literal  factor,  because  the  given 
expression  is  of  only  three  dimensions,  and  ahc  is  of 
three  dimensions. 

4.  But  there  may  be  a  numerical  factor,  m  suppose,  so 
that  we  have 

{a~\-h-^c)  (ah-\-hc-\-ca)  —  {a-\-h)  (h-\-c)  {c-\-a)  =  mahc. 

To  find  m,  put  a-=h  =  c  =  \  in  this  equation,  and  rii  =  1. 
Therefore,  the  expression  =  abc. 

2.  ^Q^o\vQa\h  —  c)-^h\c~a)-\'c\a  —  h). 

1.  For  a  =  0  this  does  not  vanish ;  hence,  a  is  not  a  fac- 
tor, and,  by  symmetry,  neither  is  h  nor  c. 

2.  Try  a  hinomial  factor ;  this  will  likely  be  of  the  form 
h  —  c]  put  h  —  c  =  0]  that  is,  h~c  in  the  given  ex- 
pression, and  there  results 

a^  {c  —  c)-\-  c^  (c  —  a)  -f  c^  (a  —  c),  which  =  0. 

Therefore,  5  —  (?  is  a  factor,  and,  by  symmetry,  c  —  a  and 
a  —  h  are  factors.  Since  the  given  expression  is  only 
of  three  dimensions,  there  can  be  no  other  literal  fac- 
tor;  but  there  may  be  a  numerical  factor,  m  suppose, 
so  that 

a^  (h-c)  +  h'  (c-a)  +  cXa  -h)  =  m  {a-h)  (b~-c)  (c-a). 


FACTORING. 


109 


col  with  respect 

« 

3t  be  one.    Put- 
hence,  «  is  a 
also  factors. 

:^ause  the  given 
and  ahc  is  of 

'm  suppose,  so 

-a)  =  mahc. 
on,  and  7/^  =  1. 


a  is  not  a  fac- 
c. 

)o  of  the  form 
le  given  ex- 

=  0. 

y ,  c~a  and 
!ssion  is  only 
r  literal  ho- 
rn suppose, 

-c)  (c~a). 


To  find  the  value  of  7n,  give  a,  b,  c  in  this  equation,  any 
values  which  will  not  reduce  either  side  to  zero ;   let 
rt  — 1,   h  =  2,   c~0,   and  we   have   2  — 7^ (—2),  or 
771  =  ~l;  so  that  the  given  expression 
—  —  {a  —  b){b  —  c)  (c  —  a),  or  (a  —  b)  (b  -  c)  (a  —^). 

3.  Resolve 

a'  {b  +  c')  -}-  P  (c  +  a")  +  c'  («  +  i')  +  cbc  (abo  +  1). 

Here  we  see  at  once  that  there  is  no  monomial  factor. 
Put  b-\-  c^  -  -  0,  that  is,  b  =  ~  c\  and    the  expression 
becomes 

a\  -c'^  +  c'O-  c\c-\-a^)--\-d'{a-^c')-d'a{-c'a-^l), 

which  =  0. 
.'.b-\-c^  is   a   factor;    and,  by  symmetry,   c-{-a^  and 

a  -{-  y^  are  also  factors ;   and  proceeding  as  in  former 

examples,  we  find  w  =  l. 
.•.  the  expression  =  (Z>  -f-  c^)  (c  +  a^)  (a  -{-  Z»^). 

4.  Resolve  into  factors  the  expression 

(a  -  bf  +  (b  -  cf  +  (c  -  a)\ 

As  before,  we  find  that  there  are  no  monomial  factors. 

Let  rt  —  Z>  —  0,  or  a  =  Z> ;  and,  substituting  b  for  a,  the 
expression  becomes  zero.  Hence,  a  -  b  is  a  factor ; 
by  symmetry,  b  —  c  and  c~  a  are  factors.  Hence, 
the  factors  are  m  {a  —  Z*)  {b  —  c)(c~  a). 

To  find  m,  let  a  =  0,  b  =  l,  c  =  2,  and  we  have 

6  =  2m,  or  7n==  3. 
The  factors  are,  therefore,  3(a~  b)(b~  c)  (c  —  a). 

5.  Resolve  into  factors 

a'ib  -c)  +  b'(c-a)  +  c'(a  -  b). 

As  before,  we  find  that  there  are  no  monomial  factors. 


no 


FACTORING. 


■ 


Let  a  —  h  0,  or  (t  h  ;  substituting  h  for  «,  the  ex- 
pression becomes  zero.  Therefore,  a~b  is  a  factor ; 
by  symmetry,  b  —  c  and  c  —a  are  factors. 

Now  the  product  of  these  three  factors  is  of  three  dimen- 
sions, while  the  expression  itself  is  of  four  dimensions. 
There  must,  therefore,  be  another  factor  of  one  dimen- 
sion. It  cannot  be  a  monomial  factor,  for  the  expres- 
sion has  no  such  factors.  It  cannot  be  a  binomial 
factor,  such  as  a-\-b,  for  then,  by  symmetry,  b-^c  and 
c-\-a  would  also  be  factors,  which  would  give  an  ex- 
pression of  six  dimensions.  It  cannot  be  a  trinomial 
factor,  unless  a,  b,  and  c  are  similarly  involved.  For 
instance,  if  a  —  b  -\-c  were  a  factor,  then,  by  sym- 
metry, b  —  c-\-a  and  c  —  a-\-b  would  also  be  factors, 
and  the  dimensions  would  be  six  instead  of  four.  The 
other  factor  must  therefore  be  a-}-b-{-c.     Hence, 

a'{b-c)-^b\c~a)-\-c'(a-b) 

—  m  (a  —  b)(b  —  c){c  —  a)(a-\-b-{-  c). 

To  find  m,  put  a  =  0,  b~\,  and  c=2,  and  we  have 
—  G  =  6m  ;  therefore,  m  =  —  \. 

Hence,  the  factors  are  —{ci  —  b){b~c){c—a){a-\-b-\  c). 
or  {a  —  b){a  —  c)  (b  —  c)  (a  -\-b-\-c). 


6.    Prove  that 

a'  +  b'  -\-  c'  +  S(a  +  b)(b  -}-  c)(c  i-  a) 

is  exactly  divisible  by  a  -f  ^  -f  c,  and  find  all  the 

factors.  , 

Let  a  +  &  -f  c  =  0,  or  a  =  —  (J  +  c) ;    substituting  this 

value  for  a,  we  have 
-(b  +  cy-{-b'-j-c'-JrSbc(b-\-c), 

or  -  (i  +  cf  +  {b  +  c)\ 

which  =  0 ;  and  therefore  a -\- b  -\- c  ia  a  factor. 


FACTORING. 


Ill 


h  for  a,  the  ex- 
i  —  h  is  a  factor  ; 
!tors. 

is  of  three  dimen- 
four  dimensions, 
or  of  one  dimen- 
r,  for  the  expres- 
t  be  a  binomial 
metry,  b-{-c  and 
uld  give  an  ex- 
)t  be  a  trinomial 
T  involved.  For 
,  then,  by  sym- 
I  also  be  factors, 
lad  of  four.  Tht- 
-c.     Hence, 

id  we  have 
—  a)(a-\-b-{  c). 


|d  find  all  the 
bstituting  this 


la  factor. 


As  bef( 


find  that  th 


l1  l'a('t( 


3  are  no  monomi 
Since  a-}- b-j-c,  the  factor  already  obtained,  is  of  one 
dimension,  the  other  factor  must  be  of  iivo  dimensions, 
and  as  it  must  be  symmetrical  with  respect  to  x,  ?/, 
mid  z,  it  must  be  of  the  form 

m  ((i^  4-  b'^  +  c"^)  -\-n  (<(b  -f-  be  -f  ca), 

•    in  which  m  and  n  are  independent  of  each  other,  and 
of  a,  b,  and  c. 

To  determine  their  values,  put  c  =  0,  so  that 
„3  _j  -h^^c'  +  3  (a  +  b)  (b  -f  e)  (c  +  a) 

-^{a-{-b-\-e)  [m  (a''  +  i'  +  <?')  4-  7i  (ab  +  he  +  ea)] 

becomes 
a'  +  ^'  +  3  ab  (a  +  b)  =  {a  +  b)  [771  (a'  +  b')  -f-  nab]. 

But 

a'  +  P  +  3ab(a  +  i)  =-  (a  +  bf. 

.-.  (a  +  by  --=  (a  +  b)  [7n  (a'  +  b')  -f  nab]. 

.-.  (a  +  by  =  vi(a'  +  b'')  +  nab. 

That 


is. 


a' 


+  b''-{-2ab  =  771  {a'  -f-  Z;^)  +  7iab. 


Now  this  is  true  for  all  values  of  a  and  b. 


m 


-  1  and  n  =  2. 


=  (a  +  ^>  +  c?)[a2  +  ^,2  -I-  c'^  _|.  2(6eZ*  +  ^'c?  +  ca)] 
=  la-\-b  -^  c)\a-]-b  -\-  cy 
=  {a-\-b~\-  cy. 


7.    Simplify 


a 


{b-\-cy+b{a-\-cy-^c{a-^by~{a^b){a-c){b-c) 
-  («  -  b){a  -  c)(b  +  e)  +  {a-b)(b~  c){a  +  c). 

Let  a  =  0,  and  the  expression  becomes 

be'  +  cb'  +  ic(i  -  c)  -  be(b  -f  (?)  -  Z»c(Z»  -  c), 

which  equals  zero ;  therefore  a  is  a  factor ;  by  sym- 
metry, b  and  c  are  also  factors. 


112 


FACTORING. 


The  expression  is  of  three  dimensions,  and  ahc  is  of 
three  dimensions,  there  cannot  therefore  he  any  other 
literal  factor. 

Hence  the  expression  equals  onahc. 

To  find  ??i,  let  a  =  i  =  c  =  1,  and  we  have 

44-4  +  4=^w;m^l2. 
.'.  the  expression  =  12 ahc. 

In  the  preceding  examples  the  factors  have  been  linear, 
but  the  principle  applies  equally  well  to  those  of  higher 
dimensions.     (See  Th.  II.  Cor.) 

8.  Examine  whether  x^  +  1  is  a  factor  of 

Let  :r'*  +  1  =  0,  or  a;**  =  —  1,  and  substituting,  the  ex- 
pression vanishes ;  therefore,  x^  +  1  is  a  factor. 

9.  Examine  whether  a^  +  h^  is  a  factor  of 

2a'  +  ceh-\'2a^V^ah\ 
Let  a^  -\- 1)^  =  0,  or  a^  =  —  V^.     Substituting,  we  have 
2h' ~  ah^  -  2h' -{- ah\ 
which  "  0,  and  therefore  o?  -f  i'^  is  a  factor. 

10.    Prove  that  a?  -\- V^  \^  i\.  factor  of 

a''-\-a'h  +  o?!)'  +  a^h"  +  ah'  +  h\ 
Let  a^  -\-P  —  0,  or  a^  -—  —  h^.     Substituting,  we  have 
-  a'  P  -  ah'  -  ^-^  +  a^  i^  +  ah'  +  h\ 
which  =  0,  and  therefore  «'  -f  ^'  is  a  factor. 


Ex.  38. 

Resolve  into  factors : 

1.  (x-\-,j  +  zy~(x^i-9/-{-z'). 

2.  hc(h  "  c)  —  ca (a  —  c)  —  ah  (h 


—  a). 


l^~ 


FArTORTNG. 


113 


,  and  ahc  is  of          ^ 

^         3. 

3re  be  any  other 

4. 

5. 

ave                           m 

6. 

-'^ 

7. 

ve  been  linear, 

8. 

hose  of  higher 

9. 

f 

10. 

11. 

tilting,  the  ex- 
3  a  factor. 

12. 
13. 

14. 

ing,  we  have 

15. 
16. 

actor. 

17. 

ng,  we  have 
Lctor. 


.r  (?/  I-  zf  +  y(2  +  .r)2  +  z{x-\-  yj  -  4.ryz. 

(.A  +  hf  -    (/>  +  cf  +  (c      «)\ 

«  (/>  -  c>^  +  /7  (r;  -  «)-^  +  c  («  -     />)\ 

{(I  -\-h  -\-  c)  (ab  -\-  he  -\-  ca)  —  nhc. 

«'  (^  +  c)  +  i'^  (^  4-  «)  +  c'(ai-  h)  +  2  «/>e. 

(a  -  -  i)  (c  -  h)  {a    -  ^-j  -f  {h  -  c)  {a  -  -  h)  (a  -  ^t) 


!)• 


18. 

19. 
20. 

21. 

22. 


{a  -  ^)^  -\-ih~  cf  +  (^  -  «)'. 

o5(6i  +  ^)  +  ^^'^(^^  +  c)  +  ca(c  +  a)  +  (n''  +  ^/  +  e^). 

a*(c  -  h')  -\-h\a-  c^)  +  c\h  -  cc")  +  ahc{a?h''c'  ~  1). 

^'  (y^  -  2^)  +  y'  {z-^  -  x')  +  z'  {x'  -  y^). 

^-'•*  +  y'  +  z'~  2xUf  -  2?/ 2'  -  2z'x'. 

(h  —  c)(x~  b)  (x  —  c)-\-(c  —  a)(x  —  c)  (x  —  a) 
-\-  (a  —  h)(x-~  a)  (x  —  h). 

+  3(a  +  2^>4-c)(Z>  +  2c  +  a)(6^  +  2a  +  ^>). 

Show  tliat  a^  +  aV^^  —  al)^  —  P  has  a^  —  b  for  a  factor. 

Sliow  that  (x  +  y)^  —  x'^  —  y'' 
=  7  :ry  (a:  +  y)  (x^  +  a.-y  +  y'y. 

Examine  whether  x^  —  5 x -{- (j  k  a  factor  of 
a^-9x'-}-2Q>x  —  24:. 

Show  that  a  —  b-\-c  is  b.  factor  of 

a'  Cb  +  c)-  P  (c  +  a)  +  c'(a  +  b)-]~  ahc. 


114 


FACTORING. 


23.  Show  that  a"^  -\-Sb  is  a  factor  of 

a' -  A  a' b' -i- S  a' h* -{- S  a' h~  12  ab*  \   9//', 
and  find  the  other  factor. 

24.  Find  the  factors  of  a*  (b     c)-\-  //*  (c  —  a)  -f-  c*  (a  —  b). 


i 


Factoring  a  Polynome  by  Trial  Divisors. 

§  26.   To  find,  if  possible,  a  rational  linear  factor  of  the 

polynome 

a;"  -f  bx""-^  +  cx""-^  + -\-  hx  -f  k 

in  which  b,  c, ,  h,  k,  are  all  integral,  substitute  succes- 
sively for  X  every  measure  (both  positive  and  negative)  of 
the  term  k,  till  one  is  found,  say  r,  that  makes  the  poly- 
nome vanish,  then  x  —  ?•  will  be  a  factor  of  the  polynome. 

Examples. 

1.    Factor  a;' +  9. i-'^ +102;  + 4. 

The  measures  of  4  are  ±1,  ±2,  and  ±  4.  Since  every 
coefficient  of  the  given  polynome  is  positive,  the  pos- 
itive measures  of  4  need  not  be  tried.  Using  the 
others,  it  will  be  found  that  —  2  makes  the  poly- 
nome vanish.     Thus, 

1         9         16        4 
-2    -14    -4 


17  2;       0 

Hence,  the  factors  are  (x  -{-  2)  (x^-{-  7.^  +  2). 

The  labor  of  substitution  may  often  be  lessened  by  ar- 
ranging the  polynome  in  ascending  powers  of  x,  and  using 
the  reciprocals  of  the  measures  of  k  instead  of  the  measures 
themselves.  Should  a  fraction  occur  during  the  course  of 
the  work,  further  trial  of  that  measure  of  Jc  will  be  needless. 


M 


i 


I 

■A 


FArTORING. 


115 


9 /A 


i-c^a-h). 


)  I  VISORS. 

^  factor  of  the 


stitute  succes- 
i  negative)  of 
akes  the  poly- 
le  polynome. 


Since  every 

sitive,  the  pos- 

.     Using  the 

:es  the  poly- 


t)- 

sened  by  ar- 
r,  and  using 
(le  measures 
course  of 
)e  needless. 


2.    Factora;''-10.c^  -63.r  +  60. 

The  measures  of  60  are  ±1,  ±2,  ±3,  dr-4,  ±5,  etc. 
Neither  -f  1  nor  - 1  will  make  the  polynome  vanish. 
Try  2 ;  thus, 

60         -63         -10         1 

30 


1 
2 


30 


16^ 


A  fraction  occurring,  we  need  go  no  further.  —  2  will 
also  give  a  fraction,  as  may  easily  be  seen.  Next 
try  3 ;  thus, 

GO         -63         -10         1 
1  20 


20         -Uh 

A  fraction  again  occurring,  v.^e  may  stop.     —  3  will 
also  give  a  fraction.     Next  try  4 ;  thus. 


60 

-63 

-10 

1 

1 

15 

-12 

4 

15 

-12 

-5i 

-  4  will  also  give  a  fraction.     Next,  trying  5,  we  find 
it  fails,  and  we  then  try  —  5  ;  thus, 


1 

60 

63 
-12 

-    10 
15 

1 
-1 

5 

12 

15 

i; 

0 

The  remainder  vanishes.     The  factors  are,  therefore, 
(a;  +  5)  (.r^- 15  ^-+12). 

§  27.  When  k  has  a  large  number  of  factors,  the  number 
that  need  actually  be  tried  can  often  be  considerably  les- 
sened by  the  following  means  : 


IIG 


FACTORING. 


For  .r  substituto  successively  three  or  more  oonsocutivo 

terms  of  the  progression  ,  3,  2,  1,  0,  —1,  —2,  -—3, 

Let ,  Z-.1,  k-i,  ki,  k,  k_i,  k_2,  ^~3, >  denote  the  correspond 

ing  vahies  of  the  polynomc  ;  and  let  r  denote  a  measure;  of 
k  positive  or  negative. 

The  substitution  of  r  for  x  need  not  be  tried  unless  r—  ] 

measure  Z',,  r— 2  measure  ki,  ,  and  also  ?• -|- 1  measuro 

k^i,  r-f-2  measure  k_2,  If  no  measure  of  k  fulfd  these 

conditions,  the  polynome  will  have  no  linear  factor. 

If  j)  denote  a  positive  or  arithmetical  measure  of  k,  the  ['receding 
criterion  may  bo  conveniently  expressed  as  follows: 

1.  Th(!  sn])stitution  of  -\-p  for  x  need  not  be  tried  unless j3—l 
iiioasuro  k^,  7)  — 2  measure  k.^, ,  and  also  j)  +  1  measure  k,^,2>'\  - 

measuro  j)_2>  

2.  The  substitution   of  — p  for  x  need  not  be  tried  unless  jj  4  1 

measure  A:,,  p  -|-  2  measure  k.^,  ,  and  also  2^  —  1  measure  /;_j,  />  —  2 

measure  k_,^,  

In  trying  for  measures,  the  signs  of  ^'2,  k^,  k,  ,  may 

evidently  be  neglected. 

If  X't  vanish,  t  positive  or  negative,  then  x--l  will  be  a 
factor  of  the  polynome,  and  should  be  divided  out  before 
proceeding  to  test  for  other  factors. 


f 


Examples. 

1 .    Find  the  factors  of  x^  —  10 .r'^  -^  63  x  +  60. 

Here  k^-60,  Z'l—  12,  k,:=~d8,  y(:_i=112,  y^_2-138. 
Tabulating  the  trial  measures,  we  get 


^; 


98 

12 

60 

112 

138 


1  '^ 

O  Q 

^,  O, 

3,  4, 
4. 


4, 
5, 


6, 
7. 


10,     12, 


FACTOR  I  N(}. 


117 


iro  fonsecutivo 

1,-2,-3, 

[he  correspond- 
,0  a  measure  of 

ed  unless  ?•--  1 
)  r-\-l  measure 
>f  k  fulfil  these 
'  factor. 

)f  k,  the  vircccding 

tried  unless  p  —  1 
iieasure  k,^,  j)  4  - 

tried  I'lnloss  p  4  1 
iieasuro  k^i,  P  ~  ^ 


ki,  /J, 


may 


- 1  will  be  a 
ided  out  before 


10. 


.12,  A;_,-138. 


^•; 


08 

12 

GO 

112 

138 


4, 

3, 
o 

1, 


4, 


7. 
6, 

5, 
4, 
3, 


C,     10, 


In  the  upper  or  positive  table,  no  measure  of  GO  gives 
a  full  column  ;  hence,  no  positive  integer  substituted 
for  X  will  make  the  given  polynome  vanish. 

In  the  lower  or  negative  table,  5  is  the  only  measure 
of  GO  that  gives  a  full  column ;  hence,  —  5  is  the 
only  negative  integer  that  need  be  tried  for  x.  Sub- 
stituting —  5  for  X,  the  polynome  vanishes;  hence, 
X  -|-  5  is  a  factor  of  x^  —  lOx"^—  63  a;  +  GO. 

In  constructing  the  above  tables  it  is  evident  that  12 
is  the  highest  measure  of  GO  that  need  be  tried  in 
the  upper  table,  for  the  next  measure,  15.,  would 
give  14  as  a  trial-measure  of  12  (the  absolute  value 
of  /:_i),  and  higher  measures  would  give  higher  trial- 
measures.  Similarly,  10  is  the  highest  measure  that 
need  be  tried  in  the  lower  table. 

Since  it  can  make  no  difference  in  the  full  columns 
which  of  the  lines  of  measures  is  made  the  basis  from 
which  to  construct  these  columns,  it  will  be  found 
best  to  construct  the  tables  by  the  measures  of  that 
one  of  the  ^''s  which  has  the  fewest  num.ber  of  them. 

2.    Find  the  factors  of  x'  +  120;^  -  40x''  +  G1  x  -  120. 

/;  =  -120,  Z;i-=-80,  ^'2  = -34,  Z;_i--238. 

Selecting  the  measures  of  34  for  trial-measures,  and 
tabulating,  we  get 


34 

80 

120 

238 


1,     2,     17,    34, 


17,     34, 
16, 
15, 
14. 


IJH 


FACTORING. 


Ilore,  in  the  only  column  that  is  full,  15  standH  in  the 
lino  of  120,  tho  ub.solutc!  viilue  of  /",  and  as  tlio  col 
umn  in  decreasing  the  sign  of  th(!  15  must  bo  minus; 
lience,  tlio  only  measure  of  k  that  need  be  tried  i.- 
-    15.     On  suljstituting  --  15  for  x,  we  get 


120 

07 

-  40 

12 

1 

-1 

8 

5 

-1 

15 

-  8 

5 

--3 

1; 

0 

Hence,  the  only  linear  factor  of  the  given  polynome  is 
X  +  15,  and,  as  is  seen  from  tho  substitution,  th(i 
other  factor  is  a;"*  —  '^x^  -\-  5a;  —  8. 


3.    Factor  x'  —  27:f''  +  14a.-  +  120. 
k  =-  120,  k,  =  108,  k,  --  50,  k  , 


80. 


56 

1,2,4,7,8,14,28,56, 

4,  7,  8,  14,  28,  56, 

108 

2,  3,    9, 

3,  6,     27, 

120 

3,  4,    10, 

2,5, 

80 

4,5, 

1,4, 

The  positive  or  increasing  columns  give  3  and  4  to  try  , 
the  negative  or  decreasing  columns  give  —2  and  —5. 
Using  these  in  order,  we  get 


120 

14 

-27 

0 

1 

1* 

40 

18 

-3  - 

-1 

3 

40 

18 

-3 

-1; 

0 

1 

10 

7 

1 

4 

10 

7 

1; 

0 

-1 

5 

-1 

o 

.J 

5 

1; 

0 

a:— 3  is  a  factor. 


07—4  is  a  factor 


a; +2  is  a  factor, 


and  there  remains  a;-f-5,  a  factor. 
Hence,  the  factors  are  {x  —  3)  {x  —  4)  (x  -f  2)  {x  +  5). 


0  Btuiid 

s  ill  tlif 

and  as 

the  ci)l 

must  l)t 

;  iniiiUH; 

eed  be 

tried  is 

ve  get 

1 

I) 

1 

1; 


0 


veil  polynome  is 
substitution,  tli(^ 


7,8, 

14, 

28,  5G, 

6, 

27, 

5, 

4, 

3  3  and  4  to  try ; 

ive  - 

-2 

and  —5. 

-3  is  a  factor. 

■—4  is  a  factor. 

|t'-|-2  is  a  factor, 

r-j-5,  a  factor. 

+  2)  (x  +  5). 


FA<;TUU1N(J. 


ll'J 


4.    Factor  :r*  —  2>r'  +  (q  —  Ijo;'  i-jKC  —  (/. 


^•  =  —  '7,  ^1 


1    .^,.|.(,^_l)a.p      ,j      0. 


Si!i(;e  botli  /•,  antl  /•  ,  vanisli,  the  polynome  is  divisible 
Ijy  both  X  —  1  and  x-j-l. 


1 

1 

7 

1 

-1 

7 

1 

■P 

-1-1 


7    i' 


(I 


+i 


)        —  <l 


0 


H« 


cnct",  the  other  factor  is  x"^  - px-\-  q. 


5.    Factor  x'  +  2r/.r"*  1-  («'^  +  (i)x' ■{-  ^a'x  f  (r\ 

Jc  ^.  a\  Ic,    -=  1  I-  2  « -f  (a'^+  a)  h  2  rr/H-  a'*  -  (>^  |  1  f 


l-2«  +  (f«2+r/,) -2 


a 


a"*  —  «•■'—  rt'^ 


a  — 


1. 


The  j)ositive  measures  of  h  are  1,  a,  cr,  (c\  Of  these  1 
may  1)0  rejected  at  once,  since  neither  Ici  nor  I:.,  van- 
ish;  and  ci^  and  a^  may  also  be  rt^ected,  since  I'l  or 
(a+  1)''  is  not  divisible  by  either  a^  ±  1  or  a^  dz  1. 
But  Z*i  is  divisil)le  by  a+  1,  and  k^i  is  divisible  by 
«  — 1;  thus,  we  need  try  the  substitution  of  only 
—  a  for  X. 


—  a 


—  a 


1 

2a 

a' 

« 

2r/2 

a^ 

—  a 

-  a^ 

-a'' 

-a^' 

1 

a 

—  a 

a 

0 

a'; 

0 

1 

0 

6t; 

0 

Hence,  the  factors  are  {x-\-(iy(x'^-\'a). 
6,    Factor  x^  —  (a  +  t*)  .r  -f  (^>  +  «6*)  a:  —  be. 


—  ic. 


1— (a+r')-f-(&-f-rte)~&c  ~  l"-a-[-h—c-{-ac—hc, 
l-(a-\-cy  -(h^~ac)-hc  =  -(l+a+^^H-(?-|-«c+6f?). 


IJU 


FA<'T(UIIN(}. 


The  lactorH  of  /•,,  otlicr  than  1,  an;  h  jiihI  c.  ky  is  nol 
divisible  by  either  h±\  nor  by  c  \-  1.  However,  /•, 
is  divisible  1)V  r  — 1,  and  h  ,  is  at  tlu^  same  tinn' 
divisi})le  by  C'\-\\  hene(\  we  wvcA  try  the  subHtitu- 
tion  ol'  only  c  lor  x. 

1  —(a +  6')  {h   \   ar)  ~hc 

c  —  ac  he 


a 


1 


a 


h 


Hence,  the  factors  are  {x  —  c)  (a;'  —  ax  -f  h). 


1.    a'-    9a^  +  16a-4. 


2. 


9a;^  +  26a;-24. 


3.  .}^'-1x''-^lQx-l2. 

4.  r*- 12a; -I- 16. 

5.  or* +  3.1;'  + 5a; +  3. 


Ex.  39. 
15. 
16. 
17. 
18. 
19. 


./;^      ll;r^  +  39a;-45. 
.r^  +  5a,-'+7.r  +  2. 
a^-3a'^-193rt+195. 
;?' -  3y  -  6;?  +  8. 
«*+3a'-3a''-7a+G. 


6.  a;*+4a;'+10a;*'+12a;+9.    20. 

7.  a;'.-3a;  +  2.  21. 

8.  a;*  +  2a;'^  +  9.  22. 

9.  m^—^7n^n-\-^mv?—27i^.    23. 

10.  a;'  +  2a;'^  +  2.  24. 

1 1 .  w?—  5  Tii^n + 8  min?~  4/^1    25. 

12.  Z>^+Z'^c  +  7ic'+39cl  26. 
1'  m*  — 47/m'  +  37i*.  27. 
14.    a*-7a''Z>+28ai'-16Z>\    28. 

29.  a;*  -  18a;' +  113a;''- 

30.  .1'*  -9a;'?/ +  29 a:' y'- 


a 


6n 


6a*'*+lla''*-0. 


a*-41«'i'  +  16Z>\ 
a*-tt'Z»''-2a^>'  +  26*. 
^9'  —  4^9'  +  6/>  —  4. 

a;'"  +  4a;'^"-5. 
y_5y3  +  8y'-8. 
a*-2a'+3a'^-2«  +  l. 
a'  +  a''Z>Ha6''-36l 
2«'»-a'"-a"  +  2. 
288  a; +252. 
-39a;?/'  +  18y. 


FACTOKINO. 


1:21 


lid  ('.     kx  is  not 

However,  /•, 

the  same  tiiin- 

ry  the  substitu- 


i^28.    To  IiikI,  il   i>t)ssil)h',  a  riilioiiul   linoar  factor  of  tlio 
I H  .ly  nome     ^^.«  .|_  /,  ,.n  i   i .  ^.^.n-a  .^ |.  }^f  j   J^^ 

ill  wlii<'li  f',  A,  c //,  k  are  all  iiitoi^Mal. 

First  Mi  (hod.     Multiply  the  polynoiiK'  l»y  <i"  '. 

or,  writing  y  for  aa:, 

if'\hif  •  h  «^2/"  '  -f h  ""  ■%  -I-  <«"  '^'. 

Factor  this  polviioiiu'   Itv  tlif  iiu'tlnjd  of  tho  last  artiulo, 
replace  y  by  ox,  and  divide  the  re.sult  by  a"  '. 


I 


Example. 


1.    Factor  ^x'  +  5:^-'  -  33a--''-f-  43  a,-      20. 

Multiply  by  3'',  and  expre.ss  in  t(M'ms  of  3^-. 

(3  x)'  +  5(3  xf  -  99  (3  xf  ■  f-  387  (3  x)  --  540 ;  or, 
y*  +  5y'    -  99/  +  387 y  --  540. 

Here  k  =  -  540,  k   ^  1  I  -"i     09  |  387-  540  ^  -  24G, 

A;  ,--=1     5  -99     387- 540 ---1030. 


24G 

540 

1030 

246 

540 

1030 


1,     2,     3,     6,    41,     82,     123,     246. 

-J,      o,      4, 


3.     6.     41, 


(Trying  by  factors  of  246, 
in.stead  of  by  factors  of 
540,  for  convenience.) 


The  only  factors  of  540  in  full  columns  are  4  in  the 
upper  table  and  2  in  the  lower  one ;  hence,  we  need 
try  only  the  substitutions  4  and  —  2. 


1 

-540 

387 
-135 

99 
63 

5 
-9 

1 
1 

4 

- 135 

03 

-9 

-1; 

0 

I 


100 


FACTORING. 


H 


onco, 


y 


4  is  a  factor.     Tho  .substitution  of  —  2  need 


not  now  1)('  tried,  since  wo  see  that  135  is  not  a  mul- 
tiple of  2.     The  other  factor  is,  therefore, 

f  V'dif      03?/-!- 135. 

Replacing  y  hy  3.r.  and  dividing  hy  27, 

■J7(3.r  -  4)  (27:i='  -I-  81.r  -  ISOn-  -!  •  135) 
^(3:^;--4)(.?;^  +  3.>:^      7^  +  5), 
which  are  the  factors. 


§J?9.  Second  Method.  Write  m.  for  "a  measure  of  a'' 
and  r  for  "a  measure  of  /;,  positive  or  negative": 

For  X  substitute  every  value  of  r-'.-m  till  one,  say  r^ -.-ru', 
be  found  which  makes  the  polynome  vanish  ;  then  'in'x  —  r' 
will  be  a  factor.  Should  a  fraction  be  met  with  in  the 
course  of  substitution,  further  trial  of  that  value  of  r-^m 
will  be  useless. 

Should  h  have  more  factors  than  <x,  it  will  generally  bo 
bettor  to  arrange  the  polynome  in  ascending  powers  of  x 
and  use  values  of  ?/?-:- r  instead  of  ?'-f-?>?,  making  r  positive 
and  w,  positive  or  negative. 

To  reduce  the  number  of  trial  measures,  for  x  substitute 
successively  three  or  more  consecutive  terms  of  the  pro- 
gression   ,  3,  2,  1,  0,  —1,    -2,  —3, ,  then  denoting  the 

corresponding  values  of  the  polynome  by  ,  Z-;,,  h>,  Z',,  ^•, 

A"_i,  A^._2,  a:__3,  

The  substitution  of  r  for  .r  need  not  be  tried  unless  r— ?/<- 

measure  /',,  /•    -2/^  measure  Z^,  ,  and  also  r-^vi  measure 

^^-1,  r4-2?/i  measure  Z;_a,  

If  j)  denote  a  positive  or  aritliTiolical  raoasure  of/;,  this  criterion 
may  be  oxjn'cssoJ  as  follows : 

1.    Tlu;  substitution  of  +jj  for  x  need  not  bo  tried  unless j)—7/i 

measure  k^^'p  —  'lm  measure  L,   ,  and  also  j)|-?/i  measure  /;  j, 

j>  !  2?u  measure  I-  ._, 


FACTORIXO. 


i23 


on  of  2  need 
J  is  not  a  mul- 
bro, 


'r, 


') 


neasnre  of  a," 
ve    : 

ne,  sav  r^ -.-rii\ 

then  7)i'x'  —  /•' 

et  with  in  tho 

-alue  of  r  -^  ?;/- 

I  generally  bo 
y  powers  of  .r 
\ing  r  positive 

|r  X  substitute 
s  of  the  pro- 
denotinn;  the 

I")  ^';i)  ^'j)  "^"ij  ^') 

II  unless  r  —  iib 

■V-'in  measure 


;,  this  criterion 

<I  unless p  —in 
\i  measure  k  ■^, 


2.  Tlio  substitution  of  —  }>  for  x  need  not  be  tridl  unless  p  +  m 
measure  h^,  p  1  2m  measure  h^,  ,   and  also  p-m  measure   ^.„ 

;)  — 2m  measure /;. 2, 

It  must  be  remembered  that  here  m  may  be  either  positive  or 
negative,  as  may  also  bi- one  or  more  of  the  quantities,  p -fm,  ^-m, 
p  i-  2w,  p-  2m,  etc. 

Examples. 

1.    Factor  3C):r='  ~\-  ITLt'^  -  22.7;  +  480. 

/;:--480,         h    -t)65,         k,        1408, 
h^,  -  C37,         Z-_,  -  920, 

and  m  may  have  any  of  the  values, 

d.1,  ±2,  ±3,  ±4,  ±G,  ±9,  ±12,  ±18,  ±3G. 

In  forming  the  table  of  trial-measures,  write  out  the 
measures  (jf  1408,  that  is,  Z-.^;  they  are 

1.  2,  4,  8,   11,   IG,  22,   32,  44,  G4,  88,  128,  17G, 
352,    1408. 

Taking  each  of  these  in  succession,  add  to  it  each  value 
of  m  separately.  Should  the  sum  appear  among 
the  measures  of  GG5,  that  is,  Z:,,  which  are 

1,   5,   7,    19,   35,   95,    133,   6G5, 

enter  these  measures  of  Tc^  and  hi  in  a  column  in  the 
table,  writing  above  them  the  value  in  used.  How- 
ever, should  the  sum  not  be  a  measure  of  G65,  another 
value  of  m  must  be  tried.  When  all  the  values  of 
'III  have  been  tried  with  one  measure  of  1408,  another 
measure  must  be  ta^cen  till  all  have  been  used.  This 
having  been  done,  proceed  to  test  which  of  the  col- 
umns can  be  filled  up  with  measures  of  480,  G37, 
and  920,  respectively,  these  being  the  values  in  this 
case  of  l\  Z' .,,  Z'_.2. 


124 


FACTORING. 


The  table  will  then  appear  thus : 
m  ;  +4,  +6,  +3,  +1.  +3.  +3,  +3. 


1408 

1, 

1, 

2, 

4,  4,  16.  32, 

665 

T). 

7, 

5, 

5,  7,  19,  35, 

k-    480 

8. 

6.  10, 

637 

7,  13 

920 

8, 

1113 

m 


-2, 

-6,  -3, 

-9, 

-3. 

-9, 

-1. 

-3, 

-9.  -4, 

-c, 

-9, 

-3, 

1408 

1. 

1.  2, 

2, 

4, 

4, 

8. 

8, 

8.  11. 

11, 

16, 

21', 

665 

-1, 

-5,  -1, 

-7, 

1, 

-r>, 

7. 

T), 

-1,  7, 

5, 

7. 

li'. 

480 

-3. 

-4. 

-16, 

-2. 

c. 

2, 

-10,  3. 

-1. 

1<!. 

637 

-7, 

-1, 

-1, 

-7, 

1.".. 

920 

-10, 

-4, 

-5. 

1<\ 

1113 

-7. 

There  still  remain  five  full  columns,  while  the  given 
polynome,  being  of  the  third  degree,  cannot  have 
more  than  three  linear  factors.  To  reduce  the  num- 
ber of  these  columns,  and,  as  a  consequence,  the 
number  of  trial-measures,  extend  the  table  by  calcu- 
lating h_z  and  the  corresponding  column-numbers 
for  the  full  columns.  ^_3  =  lil3,  and  the  column- 
numbers  are  9,  —13,  —7,  —9,  and  7.  Of  these,  9, 
—  13,  and  —9  must  be  rejected,  not  being  measures  of 
1113.   This  leaves  only  —7  and  7,  to  which  correspond 

-7—  and  -— -  as  the  values  of  —  to  be  tried  in  substi- 
2  16  r 

1 

tution  fur  -.     (See  table  above.)     Making  trial  of 

X 

these  two,  the   polynome   is   found   to  vanish  for 

--—  but  not  for  -— — 
IG  2 

The  actual  work  of  substitution  will  be  as  follows. 


FACTORTNCr. 


125 


-9,  -4,  -6. 

-9.  -3. 

8.  11.  11, 

16,  2l', 

-1.  7,  5, 

7,  1!\ 

-10.  3.  -1, 

-2,  li! 

-1.  -7. 

i:5, 

-5. 

l'\ 

<> 


while  the  givon 
3e,  cannot  hcavc 
:"e(liice  the  num- 
Dnseqiience,  the 
table  by  ealcii- 
olumn-numbers 
nd  the  columii- 
Of  these,  9, 
mg  measures  of 
lich  correspond 

tried  in  siibsti- 

aking  trial  of 
to  vanish  for 


LS  follows. 


Arrangement  in  ascending  powers  of  x : 

480  -22  171  3G 

--3  -720        1113        -1926 


o 

•J 

240 

-371 

642; 

-945 

480 

171 

36 

-3 

-  -  90 

21 

-36 

10 

30 

12; 

0 

Hence,  the  factors  are  3.r-f-16  and  12:?:'' —  7a; -f- 30. 
The  latter  factor  cannot  bo  resolved,  for  it  does  not 
contain  3.r-f  16,  and  the  only  other  factor,  viz., 
3^-|-2,  left  for  trial  by  the  tables  above,  has  been 
tried  and  has  failed. 

Factor  lO.r*  ^ x'(Wi/ -{-4:z)  -.r2(40?/-    (ji/z) 
+  .r (607/  +  IQy'z)  -  2ii/z. 

Here  iti-- dzl,  ±2,  ±  5,  or  ±  10. 
k'--'  —  24:y^z. 

h  ---  10  -  -  (15y  +  4^)  -  (40/  -  (Syz) 
+  (G0/+16y^s)-24y^2 
=  10  -15y- 40^^^4-60/ 

-2z(2-3y-8/  +  12?/') 
-  (5  -  2^)  (2  -  3y  -  8y^  +  12y'). 
^'_i-(5-h2.)(2  +  3y-8y^-12y^), 

as  may  easily  be  foimd  by  making  the  calculation. 
We  get  at  a  glance  2z  a  factor  of  X',  22;  —  5  a  factor  of 
hy,  and  22  +  5  a  factor  of  Z;_i ;    hence,  taking  m  =  5, 

we  are  directed  to  try  the  substitution  —  for  x. 

5 

10,  -  (15y+42;),  -  (40/-6y2),  (6O/+I6/2),  -24v/z 
42  -6yz  -16y'^z      24y''2 


Q, 


5 


2      -3y 


~-  8/ 


12^; 


0 


I 


12G 


FACTORING. 


TTonco,  5.r  —  '2z  is  one  factor,  tlie  other  being 

Seeking  to  determine  the  factors  of  this,  we  ohtain 
VI  =-±1  or  -±  2,  /•      12,  l\        3,     k,    =  0, 

The  vanishing  of  hi  shows  that  x  —  2'i/  is  a  factor;  and 
the  vanishing  of  k_2  shows  that  a:-f-2y  is  also  a  fac- 
tor. Dividing  these  out,  the  remaining  factor  is 
found  to  be  2a;  — 3 y;  so  that  the  proposed  poly- 
norae  resolves  into 

(5x  -  22)  (x  -  2  ?/)(.r-f  2?/)(2.r  -  3y). 

The  factor  5a;— -22;  might  easily  have  been  got  by  the 
method  of  §  23,  but  the  present  solution  shows  we  are  inde- 
})endent  of  that  section.  It  may  also  be  obtained  by  re- 
arranging the  polynome  in  terms  of  y. 


Factor  :  ^^'  '^^' 

1 .  2  x'    -  20a:-  -I-  38.r      ^0  ;  x^  -  -  7 x'^^  -\-  lGa.y  -  12i/. 

2.  12.r"'  I   r).r'^y/-h.7;/-f  3y';  Sx'-Ux  +  Q. 

3 .  3 .r^    - 1 5 ax  +  a'x-5(i^;  2 x'  -j-  9 x' y  i   7 .ry^   -  3 y\ 

4.  2Z-*^-7^;='^-4/>V^  +  ic^-46'*; 

15a'  +  47«'^^^  +  13aZ.'^-12Z/''. 

5 .  4pM-  8/  ^y  - f-  T;^^  q'  -f  8;?/^='  f  3  q\ 

6.  150.r'  -  725 .r'y  -h  931  a;^/  +  920.iy/  -  1152?/. 

7.  3Ga.'*-G(9-  7y).r' 7(9+14y).r^y+3(49--40y)a-yH-180/. 

8.  10.1'*  -a;X15y+42)  ■  |-.r(40y/H-Gy2)  f  a;(60/-lGy22)-24y'z. 

§30.    If  the   polynome    ax" -\-  bx'*~^  -\- -f- A.r  + /:,    in 

which  rt,  h,  ,  h,  I;  are  all  integral  and  7i  greater  than  3, 

have  no  rational  linear  factors,  it  may  have  rational  quad- 


1 


FACTORING. 


112^ 


jeing 


we  o1:)tain 
=  0, 

a  factor;  and 
i  is  also  a  fac- 
ling  factor  is 
roposed  poly- 


sn  got  by  tlie 
5  we  are  in  de- 
tained bv  re- 


!"  -  -  12/. 


y-    3y\ 


ly\ 


hx  +  h,  in 
ater  than  3, 
ional  quad- 


ratic Ihctors.  Lot  m  denote  a  positive  measure  of  a,  and  r 
denote  a  measure,  positive  or  negative,  of  h.  Tlie  rational 
quadratic  factors  of  the  polynomo,  if  there  be  any,  must  be 
of  the  form  rnx--\~  qx  —  r.  To  determine  such  factors  we 
may  proceed  as  follows  : 

For  X  substitute  successivelv  three  or  more  consecutive 

terms  of  the  progression  ,  3,  2,  1,  0,  —1,  —2,  -  -3,  , 

and  denote  the  corresponding  values  of  the  polynomo  by 

,  I'i,  hi,  I'l,  /i",  /'-I,  1^-2,  ^'-,i, Let  ?•;,  denote  "a  measure 

of  Z";,,  positive  or  negative";  ^'^  denote  "a  measure  of  l\, 
positive  or  negative  "  ;  etc  Then  vix'-  -\-  qx-r  need  not  be 
tried  as  a  factor  of  the  polynome,  unless  an  arithmetical 
progression  with  q  as  common  difference  can  be  formed  from 

among  the  values  of ,  9?//+ ?-3,  4?/i-f  r.2,  ^ii-\')\,  r,  ?/A-|-r  ,, 

4:/H'r>'  2,  97/^-f-y  3 ,  in  which  the  coefiicients  of  '//i,  are 

the  squares  of  the  terms  of  the  series ,  3,  2,  1,  D,  -     i, 

3, 

Examples 


o 


1. 


'»                ^7     

Factor  ./'- 

^x' 

7/i           1, 

I- 

7,               o 

h 

/•-,      -  G.^., 

h 

\^x'-\-^i\x      \H. 

--    18. 

--      G,     h    --    9, 

---102,     /:.3-        81,     Z:..4-78. 

Trying  for  rational  linear  factors  as  by  §  28,  it  will  be 
found  there  are  none.  We  therefore  proceed  to  seek 
for  rational  quadratic  factors.  To  do  this,  we  first 
tabulate  the  arithmetical  values  of  7-3,  r^,  r, 

Dm 
4  m 
III 
0         m=l 

7/t 

51,     102,    4w 

Uv/t 

30,       78.  l(i//i 


u 

'  1 

y, 

G 

*  1 

0 

3, 

G, 

3 

*  ) 

0 

18 

I) 

3, 

G, 

9, 

18 

G3 

*  1 

?>, 

H 
i, 

9, 

21, 

G3 

102 

*  ) 

0 

3, 

(>, 

17, 

31 

81 

*■} 

3. 

1^ 

27, 

81, 

78 

^  ) 

0 

3, 

G, 

13, 

2<; 

128 


FACTORING. 


Taking  these  holh  iiositivc  and  negative,  we  next  tabu- 
late the  values  of  9?M-|-r3,  4m-}-?'2,  m-\- 1\,  This 

done,  we  then  proceed  to  select  and  arrange  in  col- 
umns any  arithmetical  progressions  that  are  found  to 
run  completely  through  the  table,  one  term  of  the 
progression  in  each  line  of  the  table  in  regular  order, 
thus  : 


0 

_o 
_o 

-18 
-62 
-98 
-72 
-62 


6,      8,  10,  12,  18, 

1,      2,  3,    5,    6,    7,10, 

0.      2,  4, 

-9,   -6,  -3.-2,-1,    1,    2,   3,   6,   9,18, 

-20,   -8,  -6,-2,    0,   2,   4,    8,10,22,64, 
_47,_30,-13,-2,    1,   2,   3,   5,   6,   7,10,21,38,55,106, 

-18,      0,  G,    8,  10,12,18,36,90, 

-23,-10,  3,  10,  13,'  .,15,17,18,19,22,29,42,55,   94, 


0. 

12 

( 

4, 

n 

•  > 

8, 

— . » 

— .s 

10, 

-i;; 

12, 

-LS 

14, 

-23 

There  are  two  columns  of  progressions :  in  tlio  first,  r~(j 
and  q  or  the  common  difference  is  2,  giving  the  trial 
factor  x^  -{-2x  —  Q\  in  ohe  second  column,  ?•=— 3  and 
q-~  —  b,  thus  giving  the  trial  factor  x"^  —  5a7  +  3-  On 
actual  trial,  it  will  be  found  these  are  the  factors  of 
a;*-3^^-13:^-'H3G.r -18. 


2.    Factor  G.^^  -  53.T*  +  ll^x^  ~  299 :r'  -f  260.r  -  96. 
Here  m  may  be  1,  or  2,  or  3,  or  G. 

^_i  =  -  893. 


32, 


.*.  AXV^       .*t4» 

Vi*  \J\^ 

L  iJ      \J1 

L      »l/,       Jl/J,       J       ClrJLCi     , 

32 

"*•  > 

2, 

4,  8,  16,  32, 

9 

-^) 

3, 

9, 

4 

■*-l 

2, 

4, 

3 

^  » 

3, 

k-  96 

-*■  I 

2, 

3,  4,    6,     8,  12,  16,  24,  32,  48,  96, 

893 

1) 

19, 

47,  893, 

16/^1 
9m 
4w 

0 
m 


',  we  next  tahu- 

^  +  ^'l, This 

arrange  in  col- 
lat  are  found  to 
ne  term  of  the 
1  regular  order-, 


0,        iL- 

2.       7 

4,       2 

G,   -;; 

8,   -s 

10,  -l;: 

12,  -l.s 

14,  -2?. 


38,55,106, 
42,55,  94, 


1  tlio  first,  r~G 
iving  the  trial 
m,  r=~3and 
-5;?; +  3.  On 
the  factors  of 


X  -  96. 


l\  =  -  32, 


96. 


1 6  m 
9m 
4:m 

7)1 

0 
m 


FACTORlNr;. 


129 


I 


As  this  table  yields  no  complete  column  in  arithmetical 
progression,  the  given  pblynome  has  no  rational  linear 
factor.     (§29.) 

Forming  the  table  for  ?m— 1,  it  will  be  found  that  it 
also  does  not  yield  any  trial  divisor. 

The  table  for  m  =  2  is  : 


0,     16,     21,     28,     30,     31,  33,  34,  36,  40,  48,  61, 
9,     15,     17,     19,     22,     27, 
4,       6,       7,       9,     10,     12, 
-1.       1,       3,      5. 
_96,  -48,  -32,  -24,  -16,  -12.  -8,  -6,  -4,  -3,  -2,  -1,1,2, 3, 
-891,-45,-17,       1,       3,     21,  49,895, 


28 

19 

10 

1 

-8 

-17 


This  gives  the  complete  column  set  out  at  the  right. 
In  it,  r  =  —  8,  and  the  common  difference  is  —  9 ; 
hence,  we  have  ^x"^  ~9x-\-8  as  a  trial  divisor.  On 
actuc^  trial  it  will  be  found  to  be  a  factor  of  the  given 
polynome,  the  co-factor  being  3^;^  — 13 a;'^-}- 19a;— 12. 

Ex.  41. 

Factor : 

1.  .^•*-12ar'  +  47a;^-66.^•  +  27. 

2.  .r*-6r'-2a;2  +  36a;-24. 

3.  x*-2x'~-25x^-^18x-\-24.. 

4.  a,-5-3Lr'+1S6a;-180. 

5.  1  -  45a;2  +  32ar'  +  281  x'  -  518x^-\-2b2x\ 

6.  rt«-38a*/  +  28ay  +  345rt-^y*-564a7/5 4-180/. 

7.  2a;*-5a;^-17a:^  +  53a;-28. 

8.  6 .^'«  ~53x'  +  83  x*  -f  45  x^  -  257 :i-^  +  32 a:  +  15. 

9.  6-47?/  +  108?/^-74?/^  +  12y*. 
10.  6x^~17x*  +  bx'+lSx'~2x-2. 


CHAPTER  TV. 


Measureh  and  Multiples. 


§  31.  When  one  qnantiiy  is  to  be  divided  by  another,  the 
quotient  can  often  be  readily  obtained  by  resolving  the 
divisor  or  dividend,  or  hoth,  into  factors. 


Examples. 

1.  Divide  a'      2ah  +  b' ci'  +  2cd      (P  by  «  -  6  +  c  -  d. 

Here  we  see  at  once  that  the  dividend 
=  (a  ~  by  -~  {c  -  d)\ 
and  hence  quotient  —  a— Z>  — (tf— c?)  =  a  —  be-yd. 

2.  Divide  the  product  of  a^  -f-  ax  -f-  x^  and  a^  -\-  ^ 

by  a^+a'^x-^  +  ^*. 

Here  a^  +  r''  -—  (a  +  x)  (a^  —  ax  +  x"^),   and  the  divisor 

"  (a^  -j-  ao;  +  ^"0  (d^  —  ax  -}-  ^^)- 
Hence,  the  quotient  is  a  -|-  a;. 

3.  Divide  a!^ -\-  a^  b  -{-  d^  c  —  abc  —  b"^  c  ~  b&  by  (i^  —  be. 

The  dividend  is  a{a^  —  be)  -f-  b{ct^  —  ha)  -\'c{a^  ~  be). 
Hence  the  quotient  =^  a  +  i  +  (?. 

4.  {a""  +  ¥  ~  e^  -{-?>abe)  -^  {a  +  b  ~  c). 
Dividend^-a'  +  6''+3a^>(a+^>)  — c='~3aS(a  +  ^>)  +  3a5(7 


{a  +  by 


6-''--  '^ab(a-\-b  —  e), 


which  is  exactly  divisible  by  a  +  Z>  —  e. 
Quotient  =^  a^  -{-  b'^  -{-  c^  —  ab  -\-  be  -\-  ca. 


by  anotlier,  tlio 
resolving  tliu 


rt  — -  6  +  n  —  il 
a~h~  C'\-d. 

i  the  divisor 


a?  -  -  he. 
\c{a^~hc). 


k  +  Z')  +  3a^c 


MEASlTvES    AND    Ml'I/ri  I'l.KS. 


\?A 


5.  Divide  ./;•'    -  x' y  -\-  .r^i/  ■    .r  if"  -  \  .>;'/'      //  by  .r'      //. 

The  dividend  is  (§  124)  evidently  (^.r''      //•)  :  (.^•  j- y), 
and  this  divided  hy  x^  —  if 
-^  (r''  +  f)   :   (.t-  +  y)  --  x"-  -  -  xy  ■  I  //. 

6.  Divide  h{.x^^-Yd)-\-ax{:x'    a')-Vo\x\-(i-)  hy  (r/  K>)(.r-|  nr). 

Striking  the  factor  x-^tt  out  of  dividend  and  divisor, 
we  have  h  (x^  —  ax  -\-  a"^)  h  nx  (x  —  a)  -f  a^ 
—  h  (.r  -  ax  4-  «"0  +  (^'  ('^'^  —  cix  -f  ct^) 
=  (a  -f  />)  (.f"'^  —  ax  -\-  a'). 

Hence,  quotient  =  x^  —  ax  -f-  <r. 

7 .  Divide  ca/a?;*  -\-  .'/;''  (r< (/  - 1-  />/^)  -| "  ^^'^  {<'r-\-hq  -{-  j^c) 

Factoring  the  dividend  (§  22),  we  have 

Hence  the  quoiient  equals  the  latter  factor. 

8.  Divide  6.^*— 13rt.r''  +  13a''a,-'— 13a''a;-5a* 


by  2.f' 


Zax 


a' 


This  can  be  done  by  §  15.  The  divisor  is  2x'^--ar  —  2>ax^ 
and  we  see  at  once  that  'dx^  +  ba^  must  be  two  terms 
of  the  quotient. 

Multiplying  diagonally  into  the  first  two  terms  of  the 
divisor,  and  adding  the  products,  we  get  -\-la^x^\ 
but  +  13  a^:^''^  is  required.  Hence,  +  Ga'a;^  is  still  re- 
quired, and  as  this  must  come  from  the  third  term 
multiplied  into  —2>ax^  that  third  term  must  be  —2ax. 
Therefore,  the  quotient  is  Zx'  -|-  5a^  —  2ax. 

Note.   By  multiplying  the  terms  —2ax,  —  3a.r,  diagonally  into 
the  .c^'f^  and  a^'s,  respectively,  we  get  the  remaining  terms  of  the  div' 
dend.    It  is,  of  course,  necessary  to  test  whether  the  division  is  exact. 


I 


132  MEASURES    AND    MIT^TIPLKS. 

9.    Divi.loL>a* aV>     V2a'h'     bah^-\~W\)y  d'     iP-2>ah. 

Hero,  us  before,  one  factor  is  a^  —  1?  —  2aA  ;  hoiu;c  two 
terms  of  tlie  other  faetor  are  2a^-— 4/r.  Miilti])ly- 
ing,  as  in  tlie  last  example,  we  get  ija!^h'^\  l)ut 
—  12(t^h'^  is  required.  Hence,  —da^h'^  is  still  needed, 
and  -\-'c>ah  is  the  tliird  term  of  the  re(|uired  quotient, 
whieli  is  therefore  2a^  —  4^'^  -]-  3 ah. 


10.    Prove  that 

(1  -I-  .^•  +  iv'  +  •••••  -I  .!•"  ')  (1  -  X  +  x' 

---l\-x'  +  x''{- -f-.r^"-^ 

Product 

_  l-.i;"^  l  +  .r" 


-\-  a;"-') 


1    - 


—  X 


1  -t-  .r 


11. 


=  ¥^  -  1  +  :r^  +  .^•M^ +  ^••^"-^ 

1  —  ^^ 

Divide  (a"  ~  hcf  +  BPo^  by  a^  +  be. 

=  (a^  -  Z'c)-'  +  (2  bcf  -^  (a'  ~hc)  +  2  be 
=  (a'  -  bey  -  (a''  -  be)  x2be  +  (2bey 
=  a'~ia''be^~7b'e\ 


12.    Divide  1  +  2,357,947,69Lf«  by  1  -  ll^i-  +  12U•^ 
Dividend   =l  +  (lla;7 

=  [1  -  (11  xf  +  (ll.x-)T[l  +  (11^)']. 
Divisor       =  [1  +  (Uxf]  -f-  (1  +  11  a;). 
.•.quotient  =  [l-(lla;)^  +  (lla;)''](l-|-lla;). 


Ex.  42. 

Find  the  quotients  in  the  following  cases : 


MKASURES    ANI>    MTTT/riPLES. 


133 


'<t'     o'~2ah. 
(if> ;  hence  two 

1 '' 

'r.     Mult 

-GaV>2; 

i],ly- 

l)Ut 

1  '• 

isKtill  net 

idetl, 

1 '' 

lired  quotient, 

1 

--\-x^'-') 

\l 

1 

12L^•^ 

-       ^1 

1  ^^ 

Ixfl 

- 

1       19. 

x). 

i 

21. 
22. 
23. 
24. 
25. 

•  Ifi     I        ,8  ,„« 


'■         I"  tt    0-'       I     (i 


lA 


2   ..i 


h4. 


.2  .,2 


y 


-32 


0^,4 


?/'  - 


a   .6* 


41 


fa^ 


-y. 


1      4a,-'  +  12a;'  -  9a;*  -^- 1  +  2a;  ~  3.r'. 

(a'      2 r?a;  h  .^•■0  C^ t'  -  f-  3  a' .6-  -f  -  3  ax'  +  a/'j   :  a''  -  x\ 

./••■'      //  -f-  2'' ' h  3  xyz  ■\-x  —  y'\-z. 

Ga*      aV>  -h  2a'^i'^  +  l?>ah^  -h  4/>*  :  2r/^      ^dh  -\~  ih' 


x* 


:r^/  +  Gay--9y  :-2a;^-f3/      .nj. 


a 


+  /y 


2a-//-f- 


a 


h' 


21  ri*  -  1G«\';  +  IQxeh'  -  5aZ»''  -}  2//  :-  Za'  -ab  +  b' 


).,3  >7^,2 


L:a 


7a'~4Ga-21--2a^  +  7 


I     Q 


a  -j-  o 


[„''  (/,  -^c)  +  b'  (c  -  a)  +  c'  (a  ~  h)]  ^-  a  +  b-\-  c. 


x"      ^ax'i-3a'x-c(J'-^^-:^ 


x~  '  a 


+  i. 


X    — 


y^  H-  2 


t    I     o 


J,  X-  z 


O  ,,2 


-// 


1 


1     I         'J 


.6-'  —  (a  -f-  ^)  a;'"*  +  (b  +  ««^')  ^'"^  ~-  bcx  — 


X  —  6'. 


x'-\x^y~\-xif~\if-:-x-Y-y- 

x'  -  :i-«y  +  x^if  ^-  x'y^  +  a;'"'y*-a;^?/5_|_^y6_,^7.^_^^4_j_y 


a 


a 


a 


-\-b'  ~  a'  ~  \la?b''  -2c'    -  I 


a 


b'-c'-~l. 


-  aP  - 


«<?" 


2a'b 


O  7.4 


'/-f-2Z»6'''-l-3a'c-3Z>V-3c* 


-i-^6  +  3c'  — 2Z>. 


'  b  —  io;^  +  a'  X  —  or*  -^-  (x  +  Z>)  (a  —•  x). 
a (b  ~-  cf  -^b{n-  of  +  c(a  -  bf    -  a^  -  ab  -ac-\-  be. 
a'  1?  -\-  2  abc'  -  ce  &  -  i^  c"  -:-  a5  -|-  r;c  -  be. 
^•' +y'  + 3a,'?/ —  1 -f- a;  +  7/ —  1. 


a;'  — a;'  — 2 


a;' 


a;+l. 


1 M 


MKASrRKS    AND    MI'I/niM.KS. 


26.    a*~2da'--b0a    -'21  :  a'      fi 


^(i 


27.    (2x~f/ya*^~(x''\-i/ya'x'\'2(.r  |  ^)a.v*      j,-« 
-H-  (2a:  -    t/)  (i^  -|-  (.c  -|-  y)  ax  —  ;//'. 

-H(a?--l)rr       [.r       ])a|3. 


§  32.  The  Highest  Oommon  Factor  of  two  iilg(>l)r.'ii(!  quan 
(itios  may,  in  general,  ])e  readily  t'onnd  by  I'actoi-ing.  Tlh 
ILC.F,  is  ofttMi  (liHcovered  hy  taking  the  sum  or  the  dillci 


en(!e  ^or  snni  (OKt  dilterenee 
some  niultij)les  ui"  them 


/  dillerenee)  of  the  given  expressions,  or  ol 


Examples. 


1. 


(I'C 


FindthoII.C.F.  of(A--6')a:^-|-(2f<A      2r?r>).r  \ n'h 
and  (ah  —  ac-\-  b'^  —  he)  x  -\-  d^  c  ~\-  ah'^    -  d^  h  —  ahc. 

Taking  out  the  common  factor  Z>  —  <?,  we  get 

{h  —  c){x^~\-2ax-\-ah)  and  (h  —  c)\^a—h)x--o?-\-ah]. 

Therefore,  i  —  c  is  the  H.C.F.  of  the  given  expressions. 

2.  Find  the  H.C.F.  of  \  —  x  + ;//  -|-  z  -  xy-yyz  —  zx  -   xyz 

and  1  —  .?:  —  y  —  2:  +  xy  -|-  yz  -\-  zx  —  xyz. 

Their  difference  is 

2?/  + 2^  -  2a7y- 2237  =  2(1  —  a;)  (y  H- 2). 

Their  sum  is 

2  —  2a;  +  lyz  -  Ixyz  =  2(1  —  x){\  +  yz). 

Therefore,  the  H.C.F.  is  (1  -  x). 

3.  Find  the  H.C.F.  of  a:^  +  3  a:*  -  8a;'  -    9a;  -  3 

and  x"  -  2a,''  -  6a;'  +  4a;''  +  13a;  +  6. 

The  annexed  method  of  finding  the  H.C.F.  depends  on 
the  principle  that,  if  a  quantity  measures  two  other  quanti- 
ties, it  will  measure  any  multiple  of  their  sum  or  difference. 


If 


MKASrHKM    AND    MTn/lI  I'l.KS. 


13:1 


s^M 

'fS 

■ 

1 

(1         8 

-9 

-3 

(") 

x'*                    ^B 

1 

.2 

G        -l 

-il3 

-hO 

('') 

=„.    :„..,„    1 

2 

5 

-h6   -12 

-22 

u 

('■)  [ 

+  G 

0        10 

18 

6 

A 

1 

2 

G      -h  4 

+  13 

•16 

('') 

algcLraic  quaii       H 
liicioi'ing.     Till'      B 
III  or  tlic  dillri'       S 

3 

+  4 

-6    "12 

5 

('0 

^ 

15 

1  ]S        30 

GO 

-27 

('•)  X  ."> 

xprc'ssiuus,  or  of     ^ 

15 

-1  20        30 

-GO 

-25 

{(I)  X  5 

S 

2      -G 

G 

2 

(/') 

i 

m 

1          3 

3 

-M 

I^H 

— . 



ir. 

-1 
(r)  X  5                             i 

fr).v  \  (t^h      (t'c       ■ 

25   -1  30 

-60 

-110 

—  a'  b  —  a/;<?.         jH 

II.C.F. 

27     i  30 

-  51 

- 108 

■15 

-  h)  :v  —  a^-\-  ah  J .       fl 
en  expressions.        m 

-2     -G 

-^-6 

0 

.J 

(!/) 

1    +3 

-3 

+  1 

?/2  —  zx  -  -  :i'y~ 

17. 


-')■ 


N- 


depends  on 
other  quanti- 
or  difference. 


The  coefficients  are  written  in  two  lines,  («)  and  (^). 
They  are  then  subtracted  so  as  to  caiuiel  the  first 
terms,  {a)  is  next  multiplied  by  2,  and  added  to 
cancel  the  last  terms.  If  {c)  and  (c^)  had  been  the 
same,  their  terms  would  have  been  the  coefficients 
of  the  II.C.F.  Since  they  are  not,  we  j^roceed  with 
them  as  with  {(i)  and  (/>)  till  they  become  the  same. 
Whe^i  (a)  and  (5)  do  not  contain  the  same  number 
of  terms,  it  is  more  convenient  to  find  only  (c),  and 
then  use  this  with  the  quantity  containing  the  same 
number  of  terms.  The  general  rule  is  to  operate  on 
lines  containing  the  same,  or  nearly  the  same,  num- 
ber of  terms. 


13C) 


MEASURES    AND    MULTIPLES. 


4.     Find  thu  II.C.F.  of 

Sx^  +  -^^'  -  14a;  -I-  8  and  6a,-''  -  Ua;"'  +  13^;  —  12. 


6. 


-L'? 


-h 


14 


6   -11    +13 


+  8 
-12 


(a) 


6     +4   -28   -!  10 


15    -41    +28         (c) 
(5 -7)  (3 -4) 


(a)x2 


H.C.F.  ^-3a;-4. 


If  (a.)  and  (^)  have  a  common  factor,  its  first  term  must 
meo.sur.':  3  and  6,  and  its  last  term  must  measure  S 
and  12.  (c)  is  not,  therefore,  the  H.C.F.  Resolvo 
(c)  into  factors.  5a;—  7  is  not  a  factor  of  (a)  and  (/j). 
If,  therefore,  (a)  and  (h)  have  a  common  factor,  it  is 
3  a;  — 4.  On  trial,  3  a;  — 4  is  found  to  be  a  factor  oi 
(a),  and,  therefore,  it  is  the  H.C.F.  of  (a)  and  (h). 

5.    I?  x"^  -\- 2^^    l-  <i  si^nd  a;^  +  ?vi;  +  s  have  a  common  factor, 
prove  that  this  factor  is 

X  -] 

If  a;  — a  be  the  common  factor,  then  the  remainders,  on 
dividing  the  given  expressions  by  x  ~  a,  must  be  zero ; 
that  is,  a^  -{- pa -}-  q  =^  0, 
and  a^  +  ra  +  s  —  0,  or  (p  —  r)  a  =  s  —  q. 


Hence,  a  - 


an 


d  a;  — a 


X  — 


P 


P 


—  r 


What  value  of  a  will   make  a^'r'^  + (a  + 2)a;+ 1,  and 
a^x'^  +  a^  —  5  have  a  common  measure. 

They  cannot  have  a  monomial  factor.  Neither  can  they 
have  one  of  two  dimensions  unless  (a  +  2)  vanishes  ; 
that  is,  unless  a--  —  2,  in  which  case  the  expressions 
become  4  a;"  -]- 1  and  4a;'^  —  1,  which  have  no  common 


M 


■'M 


I 


•>. 


MEASURES    AND    MULTIPLES. 


137 


+  13.C-12. 


(a)x2 


(f 


!  first  term  must 
nust  measure  8 
I.C.F.  Resolve 
r  of  (a)  and  (/>). 
non  factor,  it  is 
>  be  a  factor  ol 
(a)  and  (h). 

common  factor, 


remainders,  on 
,  must  be  zero ; 


!/• 


^  +  -- 


2^  ~r 
-2)^^+1,  au.l 


itlier  can  they 

f  2)  vanishes  ; 

le  expressions 

^e  no  common 


factor.  Hence,  if  the  given  quantities  have  a  com- 
mon factor,  it  must  be  of  the  form  x-^-vi]  dividing 
a^x^  -\-a^  —  ^  by  x  -\-  ?>?-,  we  have  for  remainder 

5   -a^  1 

a?  1)1? -I-  a'  —  5  ---  0  or  m^  = —  ;  . ' .  m  "  -  a/(5  —  «'') , 

a  a 

in  which  -y/(5--«'^)  must  be  possible  and  integral; 

hence,  a"^  =^  4  («^  =  1  gives  values  to  ??«-  which  on 

irial  fail)  and  a  ^^  -b  2,  of  which  the  positive  value 

must  be  taken;  and,  therefore,  'Ix -\- 1  is  the  common 

factor. 

7.    Tf  the  II.C.F.  of  a  and  h  be  c,  the  L.C.M.  of 

{a  +  h)  {(c"  -  h')  and  {a  -  Z*)  (r/''  -f  IP)  is  --T— . 

Let  a  =  mc,  h  —  nc,  and  .-.  ^/^  —  7?i^c',  1/  ^  ?^^c^ 
Tlius  (a  -[-  /y)  :- :  r  (/y^  +  n) ;  ((«  —  Z»)  ---  c  {711  —  ??.), 

and  (ft'  -h  Z*-'')  :.  c'  {m'  +  n^)  ;  («•"'  -  Z.'^)   -  r'-*^  {m'  -^  77^). 
Hence,  (r/,  +  h)  («•"'  -  Z»'^)  ---=  c'  {m  +  w)  (m^  -  ^i^'), 

and    (a  -  h)  (a'  +  //)  -^  c*  (m  -  w)  ('m'  +  7i-^). 

The  H.C.F.  of  the  last  expressions  is  c^vi^  —  ?i^) ;  hence, 


6' 


8.    Tf  {x  —  a)2  measures  x^  +  ^'•^  -f-  r,  find  the  relation  be- 
tween 2'  and  ?'. 
Let  a:  -|-  m  be  the  other  fixctor ;  then 
x^  4-  qx  -f  r  =  (x  —  ay(x  -f-  ???) 
=  a:^  -f  (7^i  -  2  «)  a-'^  +  (a'  -  -  2  r/???)  .r  -j^  ma\ 

Equating  coefiicients,  ??i-   2a=0,  a'^-    2 am—  q,  mit^—r. 


Hence,  tix  =  2ti,  and 


a^    A.d^  -~  y,  2  «•''=.?', 


and  a^  ^  —  ^,  or  a"  ^  —  ^^  ;  and  a' 


r 


r 


2°'"  =  4 


Therefore,  — 
4 


3         '•        'I 

-—   01*  — \-  J— 

27'       4  ^27 


0. 


138 


MEASURES    AND    MrLTIPLES. 


Or  thus  :  Dividing  x^  -j-  qx-\-r  by  (x  —  a)'^  we  find  th( 
remainder  (^7  + 3a^)a:-|-r  —  2a',  and  as  this  will  b« 
the  same  for  all  values  of  x,  we  have,  by  equating 
coefficients, 

q  -f-  3  a'  =--  0,  and  r  -  2  a'  =  0, 
or  rf  —  27  a®  and  r^  =  4  «*' ; 


therefore  —  -\~  -t-  :  -  0,  as  before. 
4     27 


Ex.  43. 

Find  the  H.C.F.  of  the  following: 

1.  2x*  +  ^xr'-[-5x'  +  9x-S;  Sx'-2x^ -\-10x^-(jx ^^. 

2.  A''-|-(rt +  !>■'  +  («+ 1>'  +  «;  .^*•''+(^-  lK-(«-l>+«. 

3.  px^—(p—q)x^-{-(p—q)x\-q  ;  j>5.#-  (p^-5')2-'^+(jr>-|-<^).r    ^/. 

4.  «:<;'-(a~Z*):i;'-(i-c).f  -  c ;  2a.r^+(r/+2/>).f''+(^*+2d?).r  \-c. 

5.  1  --3fr-3J-:r2+lr»-.r*;  l-lJg.r-^rt'^+lyV^-'+n^ 

6.  ac'''  +  it'"'*  +  (a  +  b)c"+^ ;  a'^t''*  -f-  (x'c''  +  6'"^*"  +  5''6'^ 

7.  a\c'  +  a=^  -  2ahx'  +  ^V  -h  ^^'^^^  -  2a:'b 

and  2 aV  -  5 a V  +  3 a'' -  2 h'x*  +  5 a'b'x' -  3 a^Z/'. 

8 .  (ax -]-  %)"^  ~(a  —  h) (x  +  2;) (r^.f  +  bij)  +  (a  —  /> )' xz 

and  (rta;-  -  %)''^  -  ((f  -}-  Z>)(:i'  +  2;)(a.^  —  by)  -j-  (a  +  byxz. 

9.  a  (i'^  -  c')  +  />  (c^  -  a')  ~\-  c  (a'  -  b') 

and  a  (P  -  cr')  +  ^  (^'  -  «')  +  <^  (^*'  -  ^')- 

10.  «    -f  «2»»  -f  a"'  +  1  and  a""  -  «'"*  +  a"*  -  1. 

11 .  If  0.-'  -f"  ^'•'^'^  +  ^'^'  +  ^  ^^^^^  ^"^  +  ^'^  +  ^'  have  a  common 

factor  of  one  dimension  in  x,  it  must  be  one  of  the 
factors  of  (a  —  a')  x"^  +  (b  --  b^)x-\-  c, 

12.  Determine  the  H.C.F.  of  {a  -  bf  +  (b  -  of  +  (^  -  «)' 

and  (a'^      Z/'O'  +  {h'  -   cO'  +  (6''^  -  <<:')'. 


MEASURES    AND    MULTIPLES. 


139 


aj  we  find  th 

as  this  will  b 

ve,  by  oquatiiii 


h'x'-ZaHr. 

a-lifxz 

y)  +  {a  +  h)\vz. 

1. 

ave  a  common 
be  one  of  tho 

-  of  +  (c?  —  ^0' 


m 


13.  Find  the  H.C.  F.  of 

2(/  -  2/-  y-f-  2)x'  +  3(/  -  l)x'  -  (2i/-7/-  2y+l) 
and3(?/^-~4/  +  52/-2):i;'^ 

+  7 (y^  -  2y  +  l).r  -^  (3../  -  57/^  +  y  +  1). 

14.  If  x'^  -}'px  -\-  q  and  x"^  -\-  mx  +  7i  have  a  common  linear 


factor,  hIiow  that 


15.    Find  the  L.C.M.  orr'^^^.i^-f-Sx  -  1,   a-= 


-  X' 


—  or+l, 


a:*  —  2ar^  4-  2:^  —  1,  and 
16.    Find  the  L.C.M.  of 


X 


O   ..3 


rH-2x^-2.T+l, 


.r 


+  6.T2+ll:r-f-G,    ^^'-h  7a;^  +  14:r +  8, 


or'  +  8a.''  +  19:^'  +  12,  and  x'  +  9^:^  26a:  +  24. 

17.  Find  the  value  of  ?/ which  will  make 

2(2/'  +  y)x''  +  (lly  -  2)x  +  4  and 

^y'  +  f)^'  +  (11  y^  -  2y):t-^  +  (;/  +  5y>  +5^-1 

have  a  common  measure. 

18.  The  product  of  the  H.C.F.  and  L.C.M  of  two  quanti- 

ties is  equal  to  half  the  sum  of  their  squares  ;  one  of 
them  is  2x^  —  lla;'^  +  17^  —  6  ;  find  the  other. 

19.  If  X  +  a  and  x  —  a  are  both  measures  of 

x^  -{-px^  -\-qx  -\-r,  show  that  pq  —  r. 

20.  li  ar^  -\-  qx  +  ^'  and  a?  +  ^^  +  ^  have  a  common  meas- 

ure (x  —  of,  show  that  q^rv^  =  m^i^. 

21.  If  the  H.C.F.  of  oi^-{-px-\-  q  and  oi?-{-7nx-\-n  ha  x-\-a, 

their  L.C.M.  is 

x'^-\-  (m  —  a)  x^-\-px'^-{-  (a^-\-  mp)  x-\-a{m.  —  a)  (o^-\-p), 

22.  \{  x"^ -\- qx -\- \  QXidi  a? -\- px^ -^r  qx -\- \  have  a  common 

factor  of  the  form  .'^;-{  «,  ^how  that 
(p-iy^-ry(p-l)  +  l=--0. 


\ 


140 


FRACTIONS. 


23.  If  x^ -\- px^ -\- q  and  x^ -{- Tnx -j- 7i  have  x -{- a  for  the 

H.C.F.,  show  that  their  L.C.M.  is 
x*-\-(m  —  a  ~\-jp)  o(^  -{-p  {f)i  —  ct)  x^  -\-  a'^(a—p)x 
+  «^  (a  ~p)  (m  —  a). 

24.  If  x^  -\-px  -f  1  and  x^  -^-px^  +  3'^  +  1  have  x  —  a  for 

common  factor,  show  that  a 


a 


1-    7 

25.  Find    the   H.C.F.    of  {a}  -  Jj^f  ^- {h'  -  c^f  -\-  {c^  -  a'J 

and  a^  (6  —  c)  +  i^  {c  —  a)-\-  &  {a  —  ^). 

26.  If  a  be  the  H.  C.F.  of  h  and  c,  /?  the  H.C.F.  of  c  a^d  a, 

y  the  H.C.F.  of  a  and  i,  and  8  the  H.C.F.  ol  a,  /^, 


ar.d  (?,  then  the  L.C.M.  of  a,  5,  and  c  is 


«6g8 


97 


If  a;+<?  be  the  H.C.F.  of  the  :^'^4-«''^+^  and  x^-{-o}x-\-V , 
their  L.C.M.  will  be 
rc^  +  (a  -f-  a'  —  c)  a;'^  +  (aa'  —  c'^)  :r  +  (<^  —  c)  {a^  —  c)  c. 

28.  Show  that  the  L.C.M.  of  the  quantities  in  Exam.  2 

(solved  above)  will  be  a  complete  square  if 
X  =  y^  -{- z^  — 'if  z^. 

29.  Find   the   H.C.F.    of  x^  +  2:r«  +  3a;*  -  2a:^  +  1  and 

6a;«  +  2:^4-17a;^-7a;^-2. 

Fractions. 

§  33.  When  required  to  reduce  a  fraction  to  its  lowect 
terms,  we  can  often  apply  some  of  the  preceding  methods 
of  factoring  to  discover  the  H.C.F.  of  the  numerator  and 
denominator. 


i 
I 


■  ve, 
'■4 


Examples. 

1      ac  +  hi/-{-m/-{-hc   __  c{a-\-h)-\-y(a-{-h)  _  c-{-y 
uf-\-'Zhx  +  'lax-\-hf''fict-{-b)-{-2x[a-{-h)     f-{-2x 


cc  -i-  a  for  their 


a' 


(a~p)x 


ave  x  —  a  for  a 


C.F.  of  c  a^d  a, 

H.O.F.  ol  a,  ^, 

■    abch 

IS  ■ — — 

aySy 

md  a:^-|-a':r+^', 

I  — c)(a'  — c')c". 

es  in  Exam.  2 
lare  if 


2^:'^  +  1  and 


to  its  lowect 
iding  methods 
umerator  and 


2. 


3. 


4. 


5. 


6. 


FRACTIONS. 


141 


a 


occ'-d'h''  +  aP  __  ara-^-f^^=^--a^(«  +  ^>)] 


cc'  -  ha'  -  ah'  -{-h"        a  (a*  -  b')  -  b  (r/*  -  b') 
_  a{a{-b)(n,-^-by  _ . 


a 


(a  -  b)  {a'  -  Z/0       «'  -I-  i' 
■r'  +  xU/  +  r\?/  +  :i-^y'  +  xy*  +  / 

.'/••'^  —  .T*  ?/  +  r'  v/  —  o--"'^  ?/'  +  •'^'y*  "  y* 

Here  the  numerator  is  evidently  (x^  —  ?/)  -f-  {^  —  y),  and 
the  denominator  is 


^6  —  y6 


The  result  is  therefore 


X  -f-  ?/ 


(x  -\-  yY  —  x'  —  y'  _  hx'y  ■\-  10r\?/  -f  10  ^'^y''  +  hxy' 
{x~-{^y  +  a;*  +  y* ~  (^  +  y)* - x'y'  +  (.r^ y7^- a:y 
_  5 xy  \x^  +  y^  +  2.^•y  (a;  +  y)] 

"  (.^' + y'  +  V)  [(^'  +  y)'  +  ^y  +  ^'  +  y'  -  *'^y] 

5a;y  (:c  +  ?y)  (:r2  +  xy-\-y'')  _     5rry  (a,-  +  ?/) 


a;' 


2(a;'^  +  .ry  +  yT 
12a; +  35 


2(ar^  +  ar^  +  y^) 


ar'-lOa.'^  +  Sla;  — 30 

Here  we  see  at  once  that  the  numerator  =  (a:— 5)(a:  —  7) ; 
and  it  is  plain  tliat  a;  —  7  is  not  a  factor  of  the  denomi- 
nator ;  we  therefore  try  a:— 5  (Horner's  division),  and 
find  the  quotient  to  be  o?  --  5a:  -f  6. 

a: -7 


x^  -  -  5  .r  4-  6 


Hence,  the  result  -- 

a;*  +  2a.'^-f-9 

a-*-4a;^+8a;-21* 

The  factors  of  the  numerator  are  ra  once  seen  to  he 
a;^  +  2a;+3  and  a;'^  — 2a:-f- 3,  of  which  the  latter  is 
one  factor  of  the  denominator,  the  other  being  (Hor- 
ner's division)  a;'*  —  2a:—  7. 

Hence,  the  result  is  — ~ ~ — 

x^  -  2x  -  7 


142 


FRACTIONS. 


Ex.  44. 


Reduce  the  following  to  tlicur  lowest  terms 


1. 


2. 


3. 


4. 


5. 


6. 


.e 7.T-}-G 


x'-2x''^8x     m'       7'/-17/  +  G?/ 


X*  -f-  g V  -f  g* 


:t- 


x'  +  a;  —  12 
~-6x'  +  1x  -   3' 


ar' 


3:r  +  2.  :i;*  +  2;r''  +  9 


x'i-4:x'~5      X*      4.r'  I  4;/r'    -9 


2  -f-  bx 


2b-\-(lj''-^4c)x--2bx' 


a;='  +  2:r'^  +  2.r 

X^-}-4:X 

20.r^  +  :g^-l 

a':r  +  2aV  +  2«a;-'*  +  x* '     2dx*  +  5.r'  -  x  -  l" 


7. 


3  a 


2    .4 


2ai 


^^'+r|4--yy+2/' 


4  6fc V  -  2  a V  -  3  ax'  +  1         .  ,   A*      <^\  ,,.^,       o 


8. 


a 


:^(^>  -  g)  +  5^(c  -  g)  +  g'^fg  -  Z>) 
«i<?  (rt  —  Z>)  (^  —  c)  {c  --  a) 

9     {a-^h  +  cY 

d'{b-c)  +  h'{c~a)-{-c\a~b) 

10.  From  Exam.  4  (solved  above)  show  that 

ia~-by^{b-cy-^{c^aY  ^  (a-by-\-(b-cyHc-ay 

(a-bfi-lb-cf-Jric-af        b(a-b){h-c){c~a) 


11. 


{x  -\-  i/Y  -  -  XT'  —  ?/^ 

(x  +  y)^  -  2;'  -  y^ 


m 


FRACTIONS. 


143 


12.  Show  that 

(a    -  hy  +  (/>  -  ey  -f-  (c  ^  aY 


afl 


^  34.  Ill  reducing  complex  fractions  it  is  often  convenient 
to  multiply  both  terms  of  the  complex  fraction  by  the 
L.C.M.  of  all  the  denoininators  involved. 


i 


i 


1. 


2. 


Examples. 
bimphfy  -^^-^^ — ^ — ^^- ^• 

Here  the  L.C.M.  of  all  the  denominators  involved  is 
12 ;  hence,  multiplying  both  terms  of  the  complex 
fraction  by  12,  and  removing  brackets,  we  have 

6a: 4- 8   -8  +  Ca;__    12.i-    _    2>x 
21  ~ 


<(. 


a 


4a:   -  17 
h 


4      4a-      1 


1  -h  ^'^> 


1  + 


a\a—  b) 


1  +  ah 

Here,  multiplying  both  terms  by  1  -f-  ah,  we  get 
(^(1  +  ah)  -a-\-h  _ h(a'  +  1)  _  ^ 
l-\-ah'}~a{a-h)    '    a!'-\  l' 

1 


X  -  1  -f  - 


1 


1  + 


X 


X 


Here,  multiplying  both  terms  of  the  fraction  which 
follows  a;  —  1  by  4  —  a:,  the  given  fraction  becomes 


at  once 


X 


1  + 


4  — a: 


I 


144 


FRACTIONS. 


5. 


and  now,  multiplying  both  terms  by  4,  we  have 

4     ^  ±_^ 

4a7  — 4  +  4  — rr      bx 
It  may  be  observed  that  when  the  fraction  is  reduced 

ft  O 

to  the  form  -  -^  -,  we  may  strike  out  any  factor  com- 
b      a 

mon  to  the  two  deyiominators,  and  also  any  factor 
common  to  the  two  numerators ;  it  is  sometimes  more 
convenient  to  do  this  than  to  multiply  directly  by 
the  L.C.M.  of  all  the  denominators. 


4.    Simplify 


^a-\-h  ,  a  —  Ij" 


V  +  h' 


a'- 


Here  the  numerator  of  the  first  fraction  is  {a  +  by  + 
(a  —  hf,  and  the  denominator  is  a^  —  h^ ;  the  numera- 
tor of  second  fraction  is  {a? -\- b'^y  —  {d^  ~  b'^)\  and  the 
denominator  is  a*  —  i* ;  the  former  denominator  can- 
cels this  to  o}  -f-  i'^,  which,  of  course,  becomes  a  multi- 
plier of  the  first  numerator. 

Hence,  we  have  ^^ — '   .,  ^^)„J — ;  .,  ^,.„.,  ^J  =  ^-^A-,^-- 

{a'-]-by  —  {d'~b'y 


2d'b'' 


Occasionally,  we  at  once  discover  a  common  complex 
factor ;  strike  this  out,  and  simplify  the  result. 


a      b      c 

17x7171 

a^      h^      &      ab 


Here  the  denominator 


=/i+iY-4=ri+!+i" 


a 


,a 


1    i__r 

.a      b      c. 


and  cancelling  the  common  factor,  we  have 
-,  and,  multiplying  by  abc,  this 


abc 


abc 
be  -\-  ca  —  ab 


^ 


■^tf 


:f 


FRACTIONS. 


145 


I,  we  have 


tion  is  reduced 
my  factor  com- 

ilso  any  factor 
lometimes  more 
)Iy  directly  by 


1  is  (rt-f  Z>)2^[^ 
^  the  numera- 
-  bj,  and  the 
lominator  can- 
3omes  a  multi- 

2d'b' 
imon  complex 
result. 


ive 


Ex.  45. 


Simplify  the  following 


1     1    -i[l      i(l      ^')].     ^^  -  ^- 


2. 


l-i[l-^(l-.r)]'    «±i__iL:J 

a:      ,      a:         _I 1_^ 

.7;  +  7/     X ~y  ,     1  -  n      1  +  rr. 


rt 


a;^  ~  y' 


4 


1 


1  —  a     1  -f-  a 


3. 


1  + 


a 


!  +  «  + 


1  +  a 


^<^  +  h 


2,1' 


4. 


1  +  _L 

2a^         d'-\-b\       a      ab' 


d'  +  b'  ^  b 


2  b' 


a-\~b  .  a-~b  ,   ?    ,    b"^ 

— -—H rfc  +  6  +  -- 

c  -{-  a      c  ■  -  a  .  a 

— :}+~r:}  a  +  ^  +  7- 


6. 


32:?/2 


x—\  ,  ?/  —  1 


a: 


y 


+ 


yz  -—  za;  —  X7J 


X      y      z 


7. 


2      _2      _2      «*-f  Z,*  +  t.* 


aV;V^'^ 


5c      ac      ab 


140 


FUACTION.S. 


8. 


a^'\-a?h-\-ah''-[-P  .  a^-\-2ah  \  b' 


d'-d'h^ah'-V' 


a 


~h' 


a 


-\h 


+ 


a' 


-V  h' 


a 


.a 


a 


~  h' 


.a 


a 


(-  h      a''  +  h' 


-f 


10. 


<t 


h-\-c^ 


1-h 


h'^V 


a 


b-\-c 


Uc       J 


11. 


2(1-2:)      {l-xY      ^ 

l-\-x        (l-\-xY 


12. 


+ 
l~x 


(i+:^y 

1  +  x\ 


'X  —  rt 

.X  4-  r^ . 


+ 


'.r  +  d> 
.X  —  a  J 


I 
1 


Xj 


r 


+  1 


a/ 


.X  -\-  a  J 


■V  2  + 


'x^a^ 
jx—ai 


X 

y 


o 
O 


(I 


1+^ 

X 


X 


a 


hV 


a 


13. 


^a  f-  b/      \a  +  /j 


M  +  b, 


M 


3 


^a  -f-  b'^ 


o 
O 


^a  +  bV 


+(SIJ-' 


14. 

15. 
16. 


\a  —  bj         Kct  —  by 

x"  —  x'^  y -\- x^  if- —■  x^ 'if" -{■  xy^  —  If"  _  fx-  y^ 
x^  -\-x^x^  x^  y'^  -\-  xhf  ^  xy*  +  3^  '  V-^'  +  l/j 


1 


—  X 


i 


X 


1  -f-  X 


X' 


\1  ~x^      1  —x-\-  x^ 

Find  the  value  of 
a 


+ 


17. 


2  7ia  —  2  ?ia;      2  7ib  —  2  n:c 
Find  tlie  value  oi  ^\\--^(l  —  ^)] 

a  -  z>\^   /I  -  by 


l-\-X']-  x''      1  -f-  a;'' 


when  X  =  I  (a-\-b). 


when  x=^2\ 


i+b)   \i  +  b, 


FRACTIONS. 


14' 


18.     Kiinl  the  value  of 

■y/(a  -f  bx)  —  y/{a  —  bx) 


'lac 


biiv^n 


I 


§  35.  When  the  sum  of  several  fractions  is  to  l)o  found, 
it  is  generally  best,  instead  of  reducing  at  once  all  the 
IVactions  to  a  common  denominator,  to  take  two  (or  more) 
of  them  together,  and  combine  the  results. 


•-\-a> 


a  J 


f 


vX  —  a  J 


X 

-  x^ 


-\-h\ 


Examples. 


1.    Find  the  sum  of 


x-Vy    „    y-x    _  x'  - ?/ 


2a;-2y      2x-\-2y     x'-\-if 
Here,  taking  the  first  two  together,  we  have 

{x^yy^-{:x~yy  __x'-Vy\ 


2{x^-f) 


•2       l   ' 

X  -y 


^ if^ 

now  add  this  to .,-.—;, 

and  we  get  -^^ -. — ^ '-^-^  =  -. — -    • 

^  x"  —  y'  x^  —  7/ 


2.    Find  the  sum  of 

1  +  ^1      4u;      ,      ^x 

'T-^ — \ — :,  -f- 


x 


Here,  taking  the  first  and  the  last  together,  we  have 

{\-\-xY-{l-xY  ^    ^x     . 
1—  x^  1  —  x^ 

taking  this  result  with  the  second  fraction,  we  have 

u  +  :i-    i-W    1-^"' 

now  take  this  result  with  the  remaining  fraction,  and 
we  get 

\        16^' 


Sx 


':,+    ' 


X 


l  +  x'J      l-a^ 


?*(  '^ 


M 


MH 


FIIACTIONS. 


8. 


^.-In 


.r 


.In 


^1,+     ' 


^»  -^  1      x"  -\-  1      A"  —  1      x"  -h  1 

Taking  in  pairs  tlioso  wlio.so  donominatora  are  aliko. 


wo  llJ 


ave 


X 


Jin 


1      a: 


1 


Tho  work  is  often  made  easier  by  coynplclmg  the  divi- 
sums  represented  l)y  the  fractions. 

By  dividing  numerators  by  denominators,  this 

8^1  8  1 


-1-1-1 


_  o_ 


2a:-2  2:f-f2     2x-2     2x-\-2 

_3.'r  +  3  — a:-|-l__:^;-f-2 


2x' 


;i 


X 


X 


5.    -^ h- 


.r  —  9      X  -\-  i      X    -  8 


.'6' 


2      X  —7     X   -  1      ^'    -  6 


We  have,  by  division, 

9  9 


:t--2 


x-7 


9  9 

or z  + 


x-1 

2 


1  + 


2 


a;  —  G 


X  —  2      .r  ~  -  6      a:  —  7      a;  —  1 


_     2(2.r-8) 


2(2.r-8) 


(^-2)(.i'-G)       (:r-l)(a7-7) 

(4^  -  16)  ( L ^ 

^  ^V^''-8:i-+12      x'-~8x-{-7j 


-=  (80  -  20a,-)  -:-  (x'  -  IQx'  +  83x'  -  152a;  4-  84). 
[Denominator  =-  (x""  —  Sxf  -f  ld(x'  -  8a.-)  -f  84.] 


:itors  are  aliko, 


nlding  the  divi- 


FRACTIONS. 


140 


6.    Find  \\\o  viilnn  of  — ^-—-  H ^-.  when  x^  — -— 

By  division, 

but  tho  quantity  in  tlu;  brackets 
^  (a  +  ^).^   -4a/>  ^  0  ^^.^^^.^  .  ^  _^^  ^^)^.  ^.  4^^^^ 
(a:-2a)(:r  -2/;)  V     ^     y 

Ilonce,  the  value  of  the  given  expression  is  2. 


I 


rs.  this 


1 


-2      2a;  +  2 


x  —  6 


52a; +  84). 
+  84.] 


1. 


Ex.  46. 

SimpHfy  the  following  : 

X  -  a  I  oi^-\-  ax  -f-  a^      a?  ~  r/ 

j -. 

5  x-\-  a  x^  —  a^ 


aJ^^p         a^-^Sa^h  +  Sa/?h'      a(a-b)-~b(a-b) 
'   a'-ab4~b'~^  a'~b'  ^       a'^l-ab-\-b' 


a''^ab-\-b' 


3. 


r  1_  +  _L + _J^^  (.1 L_  ^.  _^.A 

yt  +  a;     a  —  x      a^  -\-  x^j  \a-\-x     a  —  x     a^  ■\-  x^J 


a 


+ 


ab 


+  3 


a-\-b      a  —  b      ab  —  b"^      «'•'  +  ab 
3  +  2.r      2 --3a;  ,  16a; -a;^ 


>j        X 


6. 

k 

|8. 


2  +  a;         .'r^-4 
1 


+ 


+ 


4a'(a  +  a;)      4a'(a-a;)      2a\a^-\-x') 


1  /3:i'  +  2?A  _  1  /3£ 


LV^ 


2V3a;-2yy      2V3a;  +  2y^ 
a;+l        x  —  X  1  — 3a; 


+ 


X 


+ 


2a;-l      2a;+l      a;(l-22-)      a;(4a;'-l)      a;(16a;*-l) 


+ 


9 


x  —  \ 


2a;  +  2     a;  +  2     2(.r  +  3)      (a;  +  2)(a;  +  3) 


150 


FRACTIONS. 


10. 


2{x-\-y)      2{y~x)      ^x'~y')  ,  4(..-»  +  y) 


x-y 


11.  (a-b) 


x  +  y 


+ 


2^  +  2/' 


+ 


x^ 


y* 


Jix-\-af      {x  +  hy_ 


.1.  '? 


1 


1 


a:  -f- «      a;  +  ^ 


12. 


a  +  ^*  ,     4  aa: 


+ 


+ 


8 


ccx 


a-  X 


a  —  X      a  -\-x       a  -{-  x 


a-\-  X 


) 


[a' 


a'  —  X' 


X' 


a  +  X*      (^^  +  ^' 


13     5a;-4  .   12a;+2      10^7  +  17 


14. 


9 
a 


a'  +  b 


:.+ 


lla:-8 


18 


a. 


0^3 


a^  -  b''  "^  (a  -  ^0  (»'  +  ^') 

18. 


2a^ 


ai'^ 


a 


-b* 


jg     12^+20a  ,  117a  +  28;r 


16. 


3a;4-o^      '      9a-\-2x 
4a:      17      8a,- -30  ,   10a;  -  3      5a:- 4 


X 


2x-1  ~^  2x-b 


17.    Find  the  value  of  — '       '        H '        '       , 

a-}-b-2c     a  +  b~    2d 

4cd 


when  a-}-b 


c  +  d 


18. 


19. 


20. 


21. 


X 


3n 


yn^2n 


?/" 


in    '     /V.H 


y 


3n 


r^n_yn         ^H_|_yn         ^"  _  y»         a;"  +  ^ " 

(g  -  5)-''»    _    (g  -  bf'^ 1_ 


+ 


(a-by-1      (a-by+l      {a-by-1      (a-by+1 

1 ,  1 1 

(a'-b')(x'  +  b')  "^  (b'-a'Xx'+  a')      (x'i-a')(x'+bi 

l+x   ,    l-x  2  2x'' 


1  ~  a;^      1  +  ^' 


X 


,2 


x'i-l 


oo     a'+a'b  +  ab^'i-b'  ^  (a+bY-Sab  ^  (a-bf-a'+b^ 


a3_a^i_a6'^  +  6'      ^a-byy^ab      {a-\-b) 


3     a=*-^>-' 


■  M 


If 


FRACTIONS. 


151 


§  36.  The  following  are  additional  examples  in  which  a 
know^ledge  of  factoring  and  of  the  principle  of  symmetry  is 
of  advantage. 


1. 


Examples. 
(z  -  .r) 


2  -.2 


-h? 


(x  -  7/y 

2 

X 


(2  +  y)' 

Cancelling   the   common  factor  a,'  —  y  -|-  2  in   the  two 

X  -\~  II  —  z 

terms  of  the  first  fraction,  there  .results  — — "^   ,      ; 

hence,  by  symmetry,  the  denominators  of  the  other 
two  fractions  will  be  x-^y  -Yz,  and  the  numerators 
will  be  y -\-z—x,z-\- X  —  y.  Hence,  the  sum  of  the 
three  numerators  -^x-^y  -\-  z^  and  the  result  =  1. 


2.    Simplify 


ah 


+ 


he 


+ 


ca 


(o-a)(c-b)      (a-b)(a—c)      {b-c){b-a) 

The  L.C.M.  of  denominators  is  evidently 

{a-h){h-c){c-a). 

This  gives  for  numerator  of  first  fraction  —  ah{a  —  h) ; 
and,  by  symmetry,  the  other  numerators  are 
—  hc(b-~c),  —  ca{c  —  a). 

Hence,  we  liave  -  <^b{'^-i')  +  lc(b  ~  c)  + mjc -^  a) 

(a  —  h){h~  c)  (c  —  a) 

_      (a  —  h)  (h  ~  c)(a~  c)      -, 
(a  —  h)(b  —  c)  {c  —  a) 

3.    Reduce  the  following  to  a  single  fraction : 
a  ,  h 


+ 


(«  —  h){a~  c) (x  —  a)      (b  —  a) (b  —  c) (x  —  h) 

+ £ 

(c  —  a){c  —  b)  (x  —  c) 


152 


FRAr-TTONS. 


Here  the  L.C.M.  \i>>  {a—h)(J)    c){c  -  a)(x  -a){x—h){x~c) 
the  numerator  of  the  first  fraction  is 


a 


(h  —  c)  (x  —  h)  {x  —  c), 


and,  therefore,  by  symmetry,  that  of  second  b 

—  h{c  —  a){x  —  c)  (x  —  a), 
and  that  of  tliird  is 

—  c(a  —  b)(x  ~  a)  (x  —  h)  ; 
and  their  sum  is 

—  [a  (h  —  c)  (x  —  h)  {x  —  c)  -\-h{c  —  a)  {x  —  c)  (x  —  a) 
-\-  c(a  —  h)  {x  —  a)  (x  —  h)]. 

This  vanishes  if  a  =  b\  lience,  a—  b  is  a  factor,  and 
therefore,  by  symmetry,  b  —  c  and  c  —  a  are  also 
factors.  Now  the  product  of  these  is  of  the  third 
degree,  while  the  whole  expression  rises  only  to  the 
fourth  ;  hence,  x^  ca7inot  be  involved.  The  other  fac- 
tor must  therefore  be  of  the  form  nix  -j-  n,  in  which 
m  is  a  number. 

To  determine  w,  put  a;  =  0,  and  the  expression  becomes 
abc {a  —  b-\-b~c-\-c  —  a)  =  0\  hence,  w  =  0,  or  the 
other  factor  is  mx. 

To  determine  m,  put  a  =  0,  6  =  1,  c==  —  1,  and  m  will 
be  found  to  be  1.     The  numerator  is,  therefore, 
x(a  —  b)(b--  c)  (c  —  a), 
and  the  result  is 

X 


{x  —  a)  {x 
3.    Simplify 

n-l-6 


b)  (x  —  c) 


+ 


6  +  c 


J)^{a 


c) 


(b  —  c)  (c  ~  a)      {c  —  a)  (a 

L.C.M.  of  denominators  is  (a  ~  b)  (b  —  c)  (c  —  a) ;  hence, 
first  numerator  is  a^  —  b"^,  and,  by  symmetry,  second 
numerator  is  b^  —  c^,  and  third  numerator  is  c^  —  o?  ] 
the  sum  of  these  =  0,  which  is  the  required  result. 


3. 


FRACTIONS. 


153 


4.    Reduce 


-F 


^  _2__  ^  (^v  -  yY  +  {y  ~  zf  -]  -  (z  -  :ry 


^-y      y-z      z-x  ■        {.■c-y){:y~z)i^z 
Here  the  numerator  becomes 

2{y-z){z~x)-\-2{x   ~y){z-x) 
+  2  (:r  -  3/)  (?/  -  2)  4-  {x  -  yf  -f  {y 
which  is  evidently 
[(•^*-2/)  +  0/-^)  +  (^-.r)p-0. 


.r) 


zY  -f  (2  -  x)\ 


5.    a 


■)+"(f^)" 


Observe  that  the  denominators  become  the  same  by 
changing  the  sign  between  the  fractions,  and  that 
the  expression  is  symmetrical  with  respect  to  a  and  h. 
The  niimerator  of  the  first  fraction  is 

a''  +  6  a^h'  -f  1 2  a''U'  +  8  ceh\ 

and,  by  symmetry,  that  of  the  other  is 

-  h''  -  6  hW  -  12  h\t^  -  8  h\i\ 

Their  sum  is,  therefore, 

.  =  (a«  -  h')  (a^  +  ^»«  +  6  aV/  -  8  oj'b^) 
=  (a«  -  h^)  (a'  -  ^^)2  =  {cv"  +  ^/^)  (a^  -  by, 

and  since  the  denominator  of  the  given  expression  is 
(a^  —  Z/')^,  therefore  the  result  is  c^  -^1?. 


Ex.  47. 


Simplify  the  following: 


1 .    x-  ( ^  1  +  ?/ 


^  +  y 


2.    ar^  +  ^Y+^ 


.  ^^"  ^1-  y  / 

'2a +  Z;' 


3. 


a  —  h^ 
a-\-h 


h 


4- 


a  / 


+ 


6'  +  a 


(^^^c)(6'-a)      (r-a)(a-/>)      (a-/^)(Z*-c) 


154 


FRACTIONS. 


+ 


+ 


(a—b)(a-c)      {h-a){h-c)      (c-a)(c-~b) 
h  ,  h_~_c  ,  c  —  a  I   (a  —  h)  (b  ~  c)  (c  —  a) 


-{-h      b-\-c      c-\-a      {a-\-h){b'^c){c-\-a) 


a" 


+ 


U- 


(a  +  b)  {a  -\-c)(x-{-  a)      (a  +  b){b-    c)  {x  +  b) 


(a  +  c)  (b  --  c)  {x-  +  c) 


X 


y 


8. 


(.r- 

-y)C'' 

^)' 

(«^ 

1 

-el 

+ 


(y  -  •^)  (y  ^  -  2)    (^  --  ^■)  (2;  -  y) 


+ 


(^--^OC^-^)     (t'-«)(<?-i) 


f/. 


1 


a 


\ 


10.    a;^ 


x 


3__2v/ 


r^v 


a;' 


+  2/V 


+  f 


a 


-f- 


a 


1        "^-1 


^Y 


+  2/V 


11. 


+ 


(Z>  +  c-2a)(c  +  a-2Z*)      (t?  + a- 2^>)(a  +  ^>  -  2r) 


+ 


{a-\'b-2c){b-^c  —  2a) 


12. 


Z^'^- 


+ 


c  —  a     ,    a 


+ 


(i  +  ^)"^      (c  +  6.)^      (a  +  6) 


13. 


a' 


h 


{a  ~b){a~  c)  (x  —  a)      (b  —  a)  {b  —  c)  {x  ~  b) 


+ 


(c?  —  a)  (c  —  b)  {x  —  c) 


14. 


.r(?/  +  z) 


+ 


2/  (2  4-  ^') 


+ 


z  (x  -\-  y) 


(;x  ~-  y)  (2;  -  x)      {y  ~  z)  {x  -y)      (2  -  x)  {y  -  z) 


RATIOS. 


155 


15.    {a  +  hf-\-{h~cY-\-{a-^rcY         2 i 

(a  -\-  b){b  —  c) {a  -\-  c)  a-\-  c      h  —  c  '  a-j-b 


+ 


16. 


+ 


+ 


X  {x  —  a)  (x  —  b)      a  {b  —  a)  {x  —  a)      b{b  —  a)  {x  —  b) 


Ratios. 


2 

(^- 

-y) 

.3 

{0- 

-h) 

1 

a 


§  37.  If  7  =  -,  therefore,  ad^=^  be. 
b      a 

Now,  dividing  ad  —  be  by  ca,  we  have  -  —  - 


ad  —  be  by  (?ci?,  we  have 


ad  ~  be  by  ai 


1 


a 

a 


d 


we  iiave     =  - 
b      a 


K  ^       Via  -\-nc  1 

Also,  — ; — ■ — -  =  each 


mb  -\-nd 


■n      tna  -{-7ie 
rov  — -— ^ - 

inb  4-  nd 


mt) 


of  th( 


-]~7id 


giv 


© 


en 


fract 


ions. 


(1) 

(2) 

(3) 

(4) 


a 


mb  -[-  nd 


{mh  -f-  nd)- 
mb  -\-  nd 


a         e 
b'^d 


A 


very  important  case  of  this  ism  —  l,7i  =  ±l;  hence, 
a _c  __a-\-  e a,  —  e 


b      d      b-{-d      b-d 


Al 


so 


a 


d 


a-\-  b      c  -j-  cZ 
For,  by  (2)  and  (5), 


(P) 
(6) 


a 


Or  thus 


a 


a 


a 


+  h 


a 


b      e  ~  d 


d 


-\-d 


a 


b      b 


W/  -1 


d 


a-\-b      c-{-d 


d 


a  +  i      a_^^      e_j_^      c -^  d 


d 


156 


RATIOS. 


Generally,  to  prove  that,  if  7  =  -;,  any  fraction  whose 

0      a 

numerator  and  denominator  are  homogeneous  functions  of 
a  and  h,  and  are  of  the  same  degree,  will  be  equal  to  a 
similar  fraction  formed  with  c  instead  of  a,  and  d  instead 
oih. 

Express  the  first  fraction  in  terms  of  -,  and  for  ■-  substi- 

c  00 

tute  its  equivalent  -,  and  reduce  the  result. 

Cv 

By  (2),  the  fractions  may  be  formed  of  a  and  c,  and  L 
and  d. 


jpa      c       e    ma  4- nc  4- pe      ace 
It  -  =^  -  =  -,  ' L-c—  ^^  _  or  —  or  — 

b      d     f    TYih -{- nd -\- pf      b      d     f 


raa  -|-  nc  -|-  pe 


™Kf)+"^©+^-^(j) 


iiib  -\-  '}id~\-2)J 


{,nh\-nd-\-pf)~ 

0  _a 

//lb -\~  nd -\- 2^f      ^ 

It  -—  -  and  —=*-j 
0      d  n      q 


■pc pa  =b  mc 


ona 


Tib  zt.  qd      qb  ±  nd 

•p,   ^  ma pc^ ma  ±.pc 

'  nh       qd     nb  db  qd 
pa mc  _  pa  ±  mc 


ma        pa 
— —  or  J-—  or 

nb  qb 


^y  (5). 


qb      nd       qb  ±  nd 


But 


'rna 


^,  hence  the  equality  stated  in  (8). 
nb      qb  i        ^  \  J 


h      d     f  71       q       s 

7na  ±  pc  ±:  re      pa  =b  re  =b  7ne 
7ib  ±:  qd±sf       qb  ±  sd±nf 


ma 
nb 


(V) 


(8;; 


(9) 


RATIOS. 


157 


md  d  instead 


If  an  upper  sign  be  taken  in  a  numerator,  the  corre- 
sponding upper  sign  must  be  taken  in  the  denominator;  if 
a  lower  sign,  the  corresponding  lower  sign ;  otherwise,  all 
the  signs  are  independent  of  each  other. 


1.    If? 


-,  show  that 


b      d 


The  given  fraction 


Examples. 

5a  — 4^^  5g  — 4c? 
7a +  5/)      Ic-t-^d 

b 


5^-4 


't' 


6c-^d 


0  d 


2.U^  =  %  show  that  ^^"  +  ^^3  =  -''j-^'t 
h      dH  Zd^b~\h^      Sc'd-4d' 

Dividing  the  given  fraction  by  b^,  we  have 

and  this  becomes,  on  substituting  for  j  its  equal  -- , 

o  d 

d'        d' 


3i^ 
d' 


2c'i-Sc'd 
Sc'd~-4d' 


3.    If  3a  --  2b,  find  the  value  of 
This  - 


a'-i-h' 


a 


'a' 


a 


^  +  1  j  H-  J  ^  —  y  j  [by  dividing  both  numera- 
tor and  denominator  by  i^].  But,  from  the  given 
relation 

J  =  -,  we  have,  by  substituting  for  ^, 

(A  + 1)  -^-  (I  -  f )  ---  35  ^  (-  6)  --=.  -  ^.     ■ 


158 


RATIOS. 


1*  ,  ~  -,.  pi'ove  thai  -——;.,  X  -      I  ,  1  • 


4. 


Wi         a      h      a4-b 
0  have  -  ~-  -^  — ! — ■ . 

c      d     c  -\  d 
and  til  '  mritipliod  by  -  i>;iv('S    „  — 


// 

;/"•' 


'a  -1  -  /A* 


6' 


7;J 


5.    If 


x^  -\-  aj^ 


'>x  -1- 


x^  -  a.T^  -f  i.i?  +  6'      o;"'' 


— I show  that  rp  =    • 

ax  [  o  a 


Multiplying  both  terms  of  second  fraction  by  x,  it  be- 
.?■•'  -f-  ax"^  —  bx  . 


conies 


ich  of  the  give 
difference  of  numerators 


now  each  of  the  given  fractions 


1. 


diil'erence  of  denominators      c 
Hence,  x'^  -}-  ax  ~b  —  x^  —  ax-\-b  or  2ax=^2b. 

Therefore,  x  = 


a 


6.    If  '1^<L^  =  g,  show  that  ^^  +  ":,+  '^^  ^  f  +  ''  +  ^^ 


For 


/  bd^df+fb 

ac ce  __ca  _ac  -[-  cc  -{-  ea 


bd      df    fb      bd-Vdf-^Jb 
By  (7)  making  m.  —  n~-'p  =  1. 


Aiso«^=^^-^^~-^  +  ^^  +  ^^^ 


6'^      d^    P     b''-\-d'+f 


V  By  (7). 


But 


ac      a 


J  ^      -,  hence  the  required  equality. 
od     b^ 

The  problem  is  a  particular  case  of  (9),  with  all  the 
signs  -f->  and  a  for  m,  5  for  n,  c  hr  p,  etc. 


RATIOS. 


150 


with  all  the 


if  tilt!  lViu;ti(»ns  ijircn  ficjiuil  to  olio  aiiotlnT  liiivo  ii<)t  ruonoiiiial 
terms,  mHtciul  of  H(!<!king  to  oxjircsM  tho  |»ro|)<)H(Ml  (luiiritity  in  toriiiH 
of  ono  fraction,  an<l  tlion  siihstifi.;ting  an  equivalent  fraction,  it  iH 
often  better  to  aHHUine  a  single  letter  to  reprenent  tho  common  value 
of  the  fractions  givf.'U  ('(jUiil,  and  to  work  in  terms  of  thi.<  assumed 
letter. 


7.  I 


,.    (I  }-  /> 


fj  +  c 


c-\~a 


Zi^a-^b)      4(/>-c)      b{c-~a) 
pi-ovo  that  -32a-|  35/^  +  127 1'    -0. 
Assiimo  each  of  the  given  fractions  =  .r,  so  that 

a-\-h  ^--  3  {a  -  h)x,  b-\'C-A{h   -c)x,  c-\    i  -  -  5  (c  —  a)x, 

U  Tt  O 

Hence,  adding  these  fractions,  we  have 
32«  +  35Z/  I  276?  -0. 

This  example  might  also  ho  worked  as  a  particular  case 
of  (7);  thus, 

n-{-b     ^    b-{-c    _    G-{-a 
S(a-b)      4(b-c)    ~b(c-a) 

^  20(a±h)  +  15(/;fc)  f-12(g+a)  ._  32a  +  35Z>-f27g 
C0(«     b)  +  (jO(b  -  c)  -\-  m(c  -  a)  '"  0 

Hence,  32a  +  35 Z;  +  27 c  -  0  X  - '  +  ^    =  0. 

S(a  -  b) 


8.    If^,  +  ^'=.?^/^^^_^  +  c' 


a 
b' 


r   d\b  'd-^ff 


prove  that 


'a  +  ^  +  (\^_  d'  +  c'  +  e^ 


J^  +  d-j-fJ      b'-{~d'~[-f 
Transposing  terms,  etc.,  we  have 

b'     bd  ~^  d'  ~^f    ~<y  '^'  d'  ^^'  ' 


'9^c\_ 


'-'l-dj-'^-l 


0. 


160 


RATIOS. 


That  is,  the  sum  of  two  essentially  positive  quantities 
=  0 ;  therefore  each  of  them  must  =  0.  Hence, 
we  have 


^-£=0,  and  ^-1  =  0; 


Also  ?  =  ^±4±-^;  hence,  «'  =-  f^i^Y  = 
b     b  +  d+f  '&■'     \b  +  d+/j 


therefore,  Ai+^+i'Y^ ??l±^l+£\ 
'{b  +  d+f J      b'  +  d'+f 


a 


1.    If -  =  -,  prove  that     ^^_^^.^ 


Ex.  48. 

a^  —  ah  -{- P  _  c^  ~  cd -}-  d"^ 


cd-^d' 


..n-«^|,  prove  that  «45|:^ef|J=(fijJ. 

3.  Given  the  same,  show  that  each  of  these  fractions 

\\h'-^d\ 

4.  If  2a:  =  3y,  write  down  the  value  of 

2a.-^-a:V  +  .'?/   ^^^^  ^^  x'-2>x'y  +  2y\ 


5.      If?: 


-  =  -,  show  that  J  =  — :r-^f' 

0      d      f  0      mo  —  na  —pj 

6.    From  the  same  relations  prove  that 


'a  —  mc—ne 


J)  —  ind—7if J 


RATIOS. 


161 


8.    If  -^^^ — ■ — -^  '  v\ 1  =  a,  prove  that  x  = ■  • 

g^    If^^^±ILiJ__^^E_fzi4,p,ovethat:r-^-^ 
7ix-\-a-{-  c       nx  —  h  —  a 


n  —  in 


10.    If 


a 


h  h  —  c   _    c  —  a    _     g  -f-  ^>>  +  ^ 

ay  +  ^^      ^^2  +  ca;      c?y  +  az      aa;  +  by  -\-  cz 

then  each  of  these  fractions  = 


11.    If 


a-\-b  -^  c  not  being  zero. 

a-{-b  _     b-\-c    c-\-  a 

a^b  ~  2{b~c)  ~~  3(c-a) 


x-^-y  +  z 


,  then  8a4-9^>  +  5c-=0. 


12.    If  V^^  +y(_^  _  1,  ,how  that 


a 


-X      n  -  gV^ 
a        \l+rj" 


13.    If 


a;' 


]I1^^^JL 


xz 


,^         ^          ,,         -,  and  .r,  y,  z  be  unequal,  show 
^(l-y^;)      y(l xz) 

that  each  of  these  fractions  is  equal  io  x  A^  y  A^  z. 

14.    If  ^.'  +  f"^+^  -  ^^_+|^,  show  that  each  of  these 
fractions  ■---.  (xy  —  1)  ~  (xy  —  3). 


15.    If 


25a:' -16 


^-i^,  show  that  ^=?. 
lOx  +  S         2a; -4  a; +  5      5 

16.   Ify  =  -i*£,showthat2^  +  2^+2£=2.  ■ 
b-\-c  y  —  26 


y-2c 


17.    If  Y^^  +  ^>'\      ip  +  c'V     1/^'  +  ^' 


25a'  +  27^/'^4-22c'  =  0. 


a 


)■ 


prove  that 


^. 


18.    If 


a 


^^  ~  3/2      y^  —  zx      z^  —  xy 


show  that  a^x  +  bhj  +  c'z  =  (a'  -f-  6^  +  c')  (a;  +  y  +  z). 


1G2 


11  ATI  OS. 


19.     If 


X 


=:_J^ ^ 


f 


a-\-  /j  —  c      h  -f-  c   -a      c  ■\-a—  h 

then  will  {n    -  A).^;  -[-  {h  --  r)y  -f-  {c--a)z  =■-  0. 


20.    II 


.a 


b      d 


1    then  (<l±l±^y     <^±^±JL. 


21.    ri'  ^L:^l^U  ^  ^liLth^ ...  ^'^  +  '^•^, 

show  that  (a  +  b  -}-  c)  (.i*  +  y  h  -)  ~  ota;  +  />y  -}-  cz. 


22.    If 


—     — ,  .show   that   each   ol 


a,"'  +  x\i  4-  ocd^  -\-d^        X  -|-  a 
these  expressions  =  1. 


23.    If 


K?T^)^]^(S^)''^''^''^' 


1  (a  --  b' 

6\a  +  bj      D 

unequal,  show  that  IGa^  H^  +  15  c  =  0. 


24.    If  (•^^Jl^y.A — ^,  prove  that  x^-\-9/-\-z'i-2xyz=l. 
\f/  -f  z.t7      1  -  x' 


25.    If 


a 


= = ,  show  that  a  +  6  +  c  ~  0. 

0-'  —  ?/     y  —  z     z  —  X 


26.  If  -  r:^  --,  prove  that  — ^—j  =  -^ — [         ),  ,(• 

0      a  a  —  b      ■y/{ac)  ~-y/{ba) 

27.  If  7  —  -  =  7,  then  each  is  equivalent  to  -^—^ —        ^' 

b      a     J  lb  -j-  ma  -\-  nj 

Hence,  show  tliat 

a  h  c 


2z-^2x-y     2x^2y-z      2y-\-2z-x 


when 


X 


X 


2a-\-2b-c      2b-{-2c-a      2c-^2a-b 


28.    If^  =  £.  prove  that(^J)"=^( 


c'^"  +  c^'^" 


RATIOS. 


1G3 


29.     If  — -- 


"(y  h2)      h{X'\-t)      f(.r-f7/) 


prove  that  -  (y  -  z)  -\  !  (z     x)  -f  -  (.r  —  y)  --  0. 


a 


30.    If 


a 


Lv{ny'    viz)      my{lz      /^r)      nz{mx-    ///) 


</, 


then  will  '\ (I  -  x)  H-  -  -  (m  -  y)  +  -  (n  -  2)  =  0. 


31.    112  = 


/. 


V('^/  -  aO 


?/<y 


nz 


y 


anc 


1 


y 


y/{(lX^     -    rt") 


o; 


snow 


that 


X  --- 


V(rt2'  -  «') 


32.    If 


a;'  — 


yz     y  ~  xz     V  —  xy 


1, 


"how  that  X  -\-y  \-z 


a^x  -\-  h'^y  -f-  c'^z 


a' 


+  ^H 


33.    If:^=.!^^r,and?=-.^  =  ^^1. 


a:       ?/ 


a* 


prove  that  —  +  7-,  + 


m'  I  n^  I  ^•'^  _  Q  '/^^•'^  +  ^'^  +  ^'' 


a* 


o;'^  +  y'  +  2'^ 


a 


34.    If  V' =  -=-..  then 


a'"  -  c^" 


a' 


'6'V*    (a"-£-^'+6'"y 


h      d     f  ^)''"-cf"     b''d'f-{b''-d:'^fj 


OK     T f  ^>  —  ^'^  —  "3  —        —  ^^ 
61      6,      63  b^ 

then  ^1^2  —  q2fi^3  + (-  l)"~'(?„-in^n 


_  a^^a2a:i  +  a>i^cha^  -j- 


^1  V^a^a  +  ^a  V6364  + 


164 


COMPLETE    SQUARES. 


36.    if^  +  f+g^i  +  l  +  ff, 
abc  a       0       c 


^nd.  {A-\-  B  ^  C){a  -\-h  -^  c)  =  Aa-\-  Bh  -\-  Cc, 


then  will 


A 


T,-\- 


B 


+ 


C 


l-^d'      1  +  ^'      l-{-c 


0, 


and  also r  -1 -\ r  =  0. 

a+-     6+7      c+- 


3.    Ifg.g.JandJ.J.^.l, 


A'^ 


^•' 


Complete  Squares,  Etc. 


1.   What  quantity  must  be  added  to  x^  -j-px  to  make  it  a 
complete  square  ? 

Let  r  be  the  quantity. 

Then  x"^  -{-px-{-r  =  complete  square  =  (a?  +  VO^ 
=  x'^'}-2x^  r-{-r. 

Equating  coefficients,  we  have 
2^r=p;  hence,  r=^  =  (^]' 

Or  thus :  Since  (a  -{-  xY  =  a^  -\- 2  ax -}-  x"^,  we  observe 
(see  §  12)  tha.i  four  times  the  product  of  the  extr ernes 
is  equal  to  the  square  of  the  mean ;  hence, 

4:cV  =  ^V  ;  therefore,  ^'  =  (  o )  ,  as  before. 


I 


to  make  it  a 


COMPLETE    SQUARES. 


165 


Or,  we  may  extract  the  square  root  and  equate  the 
remainder  to  zero  ;  thus  : 

x'-^-px  +  r  {x-{-^- 
x" 


2x  ■\-i—       px  +  r 


px  +  ^ 


r  — 


f 


Now,  if  the  expression  be  a  complete  square,  this  re- 
mainder must  vanish ;  hence,  we  have 

2.  Find  the  relation  connecting  a,  h,  c,  if  ax"^  -{-bx-\-  c  is  a 

complete  square. 
Assume  ax^  -f  bx  -{-  c  =  (^ct  •  x  -\-yJcY 

~  ax^  +  ^y/{ac)  x^  c. 
Now,  since  this  holds  for  all  values  of  x,  we  have 

2y/\ac)  -  h,  or  h^  =  4ac,  the  relation  required. 

3.  Determine  the  relation  amongst  a,  h,  c,  in  order  that 

aV  +  bx  -\-  be  +  b'^  may  be  a  perfect  square. 
As  in  Exam.  1,  we  have  ^a}x\bc-\-  b"^)  =  6V  ; 


hence, 


=  1. 


Ad'      b 

Or  thus  : 

Assume  aV  -|-  bx  +  bc~\-b^  =  (ax  -\-^bc  +  b'^f 
=  aV  +  2  axVbc  +  b'  -{-bc-\-  b\ 


Equating  coefficients,  we  have  b  ~  2aVbc  -f  b'^ ; 

hence,  — —  —  -  =  1    as  before. 
4:a^      b 


166 


COMPLETE    SQUARES. 


The  same  result  may  also  be  obtained  by  extracting  tlif 
square  root  and  equating  the  remainder  to  zero. 

-1.    Show   that  if  x* -}- ax^ -{- bx'^ -{- ex -{- d  be   a   complete 
square,  the  coefticients  satisfy  the  equation 
c'  -  a'd  =  0. 

Is  it  necessary  that  the  coefficients  satisfy  any  other 
equation  ? 

Extracting  thc^  square  root  of  x*  +  ^-^'^  +  ^^^  -\-  ex  -{-  d  in 
the  usual  manner,  we  have  for  the  final  remainder 


[• 


4 


.+.;-i(/.^| 


2\2 


Now,  if  the  expression  be  a  complete  square,  this  re- 
mainder must  vanish  ;  and,  that  it  may  vanish  for 
general  values  of  x,  we  must  have 


«/4 


^^i^-" 


a' 


Eliminating  b ,  we  have  c^ 


a'd  =  0 


(1) 
(2) 
(3) 


The  coefficients  must  satisfy  the  equations  (1)  and  (2), 
and  therefore  either  of  these  equations,  together  with 
the  equation  (3),  which  results  from  them. 

The  same  result  may  be  obtained  by  assuming 
X*  +  ax^  +  bx"^  -]-  cxi-  d=^  (x"^  -{-  ^ax  -{-y/df 
—  a;*  -j-  ax^  -f  2  x"^  -y/d  +  J  a^x"^  +  ax-y/d  -f  d. 

Equating  coefficients,  we  have  2^yd-\~\a'^  =  h  (1) 

and  a^d—c  (2) 

From  (2)  we  have  c^  —  a^d  =  0,  a^  before. 


;1 

I 


fOMI'IvKTK    .S(.iUARE.S. 


1G7 


What  must  be  tlie  Vcalue  of  m  and  of  n 

if  4:X*  —  12. r^  +  2bx'  —  4:mx-\-8fi  is  a  perfect  square  ? 

Assume  the  expression  —  [(2x^  -dx -{--^(8n)Y 

=  ix*  -  12.r'  +  4.f'  V(^^0  +  ''^'^•'  -  C.^V(8n)  -f  8  n. 

Equating  eoelhcients,  we  have  0-^/(8  ?^)  —  4i7n  [1) 

and  4V(8n)  +  9-25  (2) 

Therefore,  ?i  -  2,  ?h  —  6. 

Or  thus  :    Extracting  tlie  square  root  in  the  ordinary 
way,  the  remainder  is  found  to  he 
(—  4:  VI  +  24)^'  -{-  8?^  —  10  ;  hence,  we  must  have 
4 7)1  +  24  -=  0,  or  ra  =  (j,  and  871  —  16  =r^  0,  or  71 


o 


6.    If  ax^  -\-  hx"^  -\-  ex  -\-  d  be   a  complete   cube,   show   tliat 
ac^  =  db^  and  /^^  =:^  3  ac. 

Assume  ax^  +  ^^^  -\-  cx-{-  d=^  {ct^x  -\-  d^y 

=  ax'  +  3  aklh'  +  3  ahUx  +  ri. 
Equating  coeflicients,  b  --  3c/^(:/5 

c  =  3«J(:Zt 


Dividing  (1)  by  (2),     -^  -77  ;  hence,  «c^  --=  c^^'' 

c      as 


(1) 


Also, 


Z^'^-Qa^^l 


(3) 


Dividing  (3)  by  (2),    —  =  Sa;  hence,  b^  ==  Sac. 


7.    Find  the  relations  subsisting  between  a,  b,  r,  d,  e,  when 
ax*  -f-  bx^  -{-  eo;'^  -f  <^^^  +  c  is  a  complete /o?^r/f A  power. 

Assume  ax*  +  bx^  +  c.'^''^  +  c/.r  +  c  ^-  (air  +  c^)* 
=  a.r*  +  4  ahkx^  +  6  a^^?^-'  +  4  akix  +  e. 

Equating  coefficients,  we  have 


whence, 


b==- 

--  4  a4C4, 

c  = 

:  6  asci, 

d^- 

^  4aM; 

bd= 

:  IGae 

bCr=. 

:24aM 

cd~ 

-  24  alel 

6cx4aW  =  'i6c 


(1) 

(2) 

(3) 


168 


COMPLETE    SQUARES. 


R.  Show  I'hat  a;*+2Ar'' -j  r^a:'^ -}-7U'-|- 8  can  ^jc  resolve'"!  iri^o 
two  rational  quadratic  factors  if  s  bo  a  perfeci  square, 
negative,  and  equal  to 


p^  —  ^q 

Since  —  .<?  is  a  perfect  square,  let  it  be  v?. 

Assume  x^  -\-px^  -\-  qx^  +  ^'^  ~  'f^ 
=  (^c^  +  mx  +  n)  (x^  -\-  vi'x  —  n) 
=  x^  -}-  (m  +  w')  a^  -\-  inin^x^  —  n  {im  —  m^)x-  t^. 

Equating  coefficients,  we  have 


nimi^  =  q, 


m  —  i7i 


r 
— > 
n 


4wm'  --  4  2'. 
Hence,  (?^>t  —  7)i'y^  =p'^  - 


Therefore, 


J. 


4^y^ 


ri^  — 


•s. 


Ex,   ±t^. 


1.  What  is  the  condition  that  (a  —  x)  (o  —  x)  —  c^  may  be 

a  perfect  square  ? 

2.  Find  the  value  of  'n  which  will  make  2x'  -^-Sx  -{-n  a 

perfect  square. 

3.  Find  a  value  of  x  which  will  make 

X*  -{-  6x^-{-  ll.r^  +  3.^-1-31  a  perfect  square. 

4.  E.^ract  the  square  root  of 

(a  -  by  -  2  (a'  +  b')  (a  -5)^  +  2  (a*  +  b*). 

5«    Find  the  values  of  tn  and  n  which  will  make 
4a;*  —  4  r"*  -{-  bx"^  —  mx  +  w  a  perfect  square. 


COMPLETE    SQUARES. 


]  09 


6.  What  must  be  added  to  x'--^(4x*-1(jx'-{-1(j)'  Ax' 

in  order  to  make  it  a  complete  square  ? 

7.  Tlie  expression  x*  -f  x^—  IQx"^  —  4.r  -f  48  is  resolvable 

into    two    factors   of    the    form   x^  -j-  /hx  -\-  G    and 
x'  -\'  nx  -f-  8.     Determine  the  factors. 

8.  Find  the  value  of  c  which  will  make 

4  X*  —  coi^  -{-  b  x"^  -f  -^  +  1  a  complete  square. 

•J 

9.  Obtain  the  square  root  of 

10.  If  (a  -  h)  x'  +  (a  +  Ifx  -\-  (rr  -  b')  {a  +  b)  be  a  com- 

plete square,  then  a  ---  3Z>,  or  b  =  3a. 

11.  Find    the   simplest  quantity  which,   subtracted    from 

a'V  -j- 4  abx -\-  4  acx  -\-5bc-\-  b'^c',  will  give  for  remain- 
der an  exact  square. 

12.  .r* -— 4:1'^  —  :?;'^ -]- 16a;— 12  is  resolvable  into  quadratic 

factors  of  the  form  x"^ -{- 7nx -]- p  and  x'^ -{- 7ix -\- q  . 
find  them. 

13.  Find  the  values  of  7n  which  will  make  x'  -[- 7nax -{- .  r 

a  factor  of  x*  —  ax^  +  a'V  —  a^x  +  n*- 

14.  Show  that  if  x*-{-  ax^-}-bx^-\-cx-^d  be  a  perfect  square, 

the  coefficients  satisfy  the  relations  8  c  =  a  (4  b      a'^) 
and  64  d  -  (4  i  -  a'f. 

Investigate  the  relations  between  the  coefficients  in 
order  that  ax'^  +  ^!/^  +  cz^  +  (^^'1/  +  cj/z  +/^'2  niay  be 
a  complete  square. 

If  a^  -]-  ax"^  +  ^^^  +  c  be  exactly  divisible  by  (x  -f-  dy\ 

sliow  that  i  (b'  -  d')  -=  -,  -  d(a  -  2^?). 

d 

17.    Determine  the  relations  among  a,  h,  c,  d,  when 

ax^      bx'  -f  c^  —  d  is  a  complete  cube. 


15. 


16. 


170 


COMPLETE    SQUARES. 


18.  The  polyricmie  ax^ -\-3hx'^ -\^- Sex -\- d  is  exactly  divis- 

ible by  (a  —  xy  ;  sbow  that 
(ad  -  bcf  -  4  (ac  -  I/)  (hd  -  c'). 

19.  Find  the  reLT^tion  between^?  and  y  when  .i''-f^a;''-|- </ 

is  exactly  divisible  by  (x  —  ay\ 

20.  If  x'^ '-{-  nax  -\-  ct}  be  a  factor  of  x^-\-  ax'^-\-  aV+  (^^-^  +  «*, 

show  that  n^  —  w  —  1  =  0. 

21.  If  ./;*  -[-  (ix^  -f-  o.?;'^  -|~  <^-^'  "h  '^^  ^^  ^^^^  product  of  two  coni- 

}»lote  squares,  show  that 

22.  Prove  that  :<;*  -\-px^  \-  gx'^  +  r:f  +  s  is  a  perfect  square 

if  jP^.s  -^  r"^  and  (/  --^  ~-  +  2->/.s'. 

23.  If  a.!-^  -f  3 hx^  -j-Scx-'rd  contain  ax'^  -[-  2 hx -f-  c  as  a  fac- 

tor, the  former  will  be  a  complete  cube,  and  the 
latter  a  complete  square. 

24.  li  7n^x'^ -\- 2')x -^l- X''l '\~  T  ^^  ^  perfect  square,  find  p  in 

terms  of  rn  and  q. 

25.  Find  the  relation  between  j9  and  q  in  order  that 

x^  -{-r^x"'  -f-  qx  -^\-  f  nriay  contain  {x  -j-  2)'^  as  a  factor. 

26.  If  A^  +  /^r'^  -f  qx  -f  ?'  be  algebraically  divisible  by 

Zx'  f  2y.a'  +  3')  show  that  the  quotient  is  a:  +%• 

o 


■  ■''*<(!?.(.  ;trr**  J 


iiutly  divis- 


)f  two  com- 
'feet  square 


c  as  a  fac- 
3e,  and  the 

3,  find  p  in 

that 
s  a  factor. 

)le  by 
^3 


CPIAPTER  V. 

Linear  Equations  of  One  Unknown  Quantity. 

§  38.  Preliminary  Equations.  Although  the  follow- 
ing exercise  belongs  in  theory  to  this  chapter,  in  practice 
the  numerical  examples  should  immediately  follow  Exer- 
cise I.,  and  the  literal  examples  Exercise  III.  Like  those 
exercises,  this  one  is  merely  a  specimen  of  what  the  teacher 
should  give  till  his  pupils  have  thoroughly  mastered  this 
preliminary  work.  But  few  numerical  examples  are  given, 
it  being  left  to  the  teacher  to  supply  these. 

Ex.  60. 

What  values  must  x  have  that  the  following  equations 
may  be  true  ? 

1.  X  -b-^-0\  a;-3J  =  0;  x  —  a^O\  ar4-3  =  0. 

2.  a;  +  4|=0;:c-|-a  =  0;  .r  +  3=.5;a:-4  =  6. 

3.  x  —'  a^=^h  \  X  -{'  a  =  c;  x  —  b  =  —  c;  G  —  a;  =  3. 

4.  8-07=10;   S  +  .r^ll;   9  +  x  =  i;   7~x  =  -b, 

5.  8 -{-a;  =  —  6;  a~x~Sh;  2«  =  a;  +  3Z>;  Sa  =  bb—x. 

6.  2.r-G-=8;  3a:  +  8  =  20;  a;r  =  a^  mx  =  hn. 

7.  3  a:  =  c ;  a.^r  =  5  ;  ax  =  0;  (a-{-h)x-~  h  -\-  a. 

8.  (a~b)x^b-a',  (a+b)x-=(a+by;  (a~-b)x=a''-b\ 

9.  (a  f-  b)x  =  b''  -  a'  ■   (a'      ub  +  b'')x  -=  a'  +  b\ 

10.    (a' -  b')x  ■-=  a      b  ;  (a' -~- P)x :- a -[- b  ;  (a'^  +  Z.>   ^.l. 


172 


LINEAR   EQUATIONS. 


11.    a-\'  X  ~h  —  a  +  b  ;  x  —  a-\-b  =  b  —  x-\- 


a. 


12.    2a  —  x=^x--2b 


ax 


+  bx 


ax 


ex. 


13.    ax  —  b  =  bx--c:  ax  —  ah 


ac. 


14.    ax 


a 


=  bx—-b^\  ax  —  a^  ■=  bx  —  ¥ 


15.  ax  —  a^  —  h^-bx]    ax  -{-  b  -\-  c  —  a  -[-  bx  -[-  ex. 

16.  a~bx  —  c  =  b  —  ax-\-ex\   a-^-hx-^-ex^^^ax—b-^ 


ex 


17.    hx  —  coi^-{-e=:^ex  —  b 


ex 


'X 


i;  4 


X 


I- 


a 


18.  10a:  =  -  — 1;   ax^=-\    ax^= 

6  c  b 

19.  a6a:=:--f-;    ocx  —  '—--\ 

b      a  be 

20.  |a;  =  5;   |:r  =  8;   0.5:r  =  2;   0.3.r  =  0.06. 

21.  0.02a;  =  20;   0.3a;  =  0.2;   0.4a:  =0.6. 


22. 

0.18ar-1.8;   "^-b-    ^f - c, 
a             b 

23. 

ax      h         X                  ax        i 

7              '      1                    7            *-■    '                     7            ^* 

6        C       61  +  0               «-f-o 

24. 

a-\-b         a,     a  —  b         aArb 
-  ^  ^x-     ;   ^a;-— i— . 

a  —  0         0     a-f-  0         0  —  a 

25. 

a                a        b  —  a         a  —  b 

X  —               '                 X  — ' 

b  —  a        a  —  b^  a-\-b         b -\- a 

26. 

a-\-h         a-e     112      3 
00  —  — -— ; ; r- 

a-{-  c 


a 


+  h 


X 


X 


27.    - 


1       1 


a      a 


X 


ah 


X 


X 


b,   7^1     1 
c'    X     3     4' 


28. 


+ 


33 


a 


20      bx      5 


X 


X 


+  2  =  0. 


29. 


=  6 


X 


X 


7'    3 


X 


7  + 


4-3 


X 


LINEAR    EQUATIONS. 


1  »^o 


30.  (x-A)     (^-f-5j4-a;  =  3;  2a:  -(a:- 5)-- (4- 3^') --5, 

31.  2(3-a:)+3(a:-3)r^0;2(3a;-4)    3(3~4.r)  KX2-a,-)-10. 

32.  a (1  --  2.r)  -  (2a;  —  a)  —  1 ;  x  —  b (a  -  x)  —  hx  —  ba. 

33.  ?Aia;(3a  -4)  -}  3  wi.r  —  3 a  +  1  =  0. 

34.  a  {hx  ~c)-\-b(cx  —  a)  -\-  c  (ax  —  b)~0. 

35.  a  (ax  —  b)-[-b  (ex    -  c) -^  c  (ex  —  a)  -~  0. 

36.  a  (hx  —  a)  -{-b  (ax  —  b) -^  c  (ax  —  c)  =  0. 

37.  a(x  --2b)-\-b(x~  2c)  -f  c(x  -  2r0  =  a'  -[-  V'  +  c\ 

38.  353[3(3a;--2)-2]   -2S-2  =  L 

39.  9J7[5(3.'r-2)-4]-6|--8--l. 

40.  .Ui[i(4[^  +  2]  +  2)  +  2]  +  2J  =  l. 

41.  ll|[iab  +  2]  +  4)  +  G]  +  8i  =  l. 

42.  i-JKia^-i)-i]-il-4  =  o. 

43.  IM[t(|^-H)-ii]-HS-H--=o. 

44.  ||[A[f(|[|a:  +  4]  +  8)  +  12]  +  20J  +  32-58. 

45.  IS|[|-a[^-  +  7]-3)  +  6]-li  =  4. 

46.  ?•  J ^ [p  (ri [?72a;  —  a]  —  Z>)  —  <?]  —  c?}  —  e  =  0. 

47.  (l  +  6:r)'^  +  (2  +  8a;)^  =  (l  +  102:)^ 

48.  9(2a;-7y4-(4rr-27)'^:^13(4:u+15)(a;+G). 

49.  (3  -  4 xf  +  (4  -  4 xf  =:  2 (5  +  4 x)\ 

50.  (9  -  4x)  (9  -  5.t)  +  4(5  -  x) (5  -  4a:)  =  36(2  -  x)\ 

§  39,  111  solving  fractional  equations,  the  principles  illus- 
trated in  the  sections  on  fractions  may  frequently  he  applied 
with  advantage,  as  in  the  following  cases. 

When  an  equation  involves  several  fractions,  we  may 
take  two  or  more  of  them  together. 


1V4 


LINEAK   KQUATIONS. 


Examples. 

1.     Solve  '  --  ~ ■ — • 

14  (')./•  hii  7 

IIci'o,  instead  of  multiplying  tlirongli  by  the  L.C.M.  of 
the  (lonominators,  we  combine  tlin  first  fraction  witli 
the  last,  getting  at  once 

7a; -3       7       1 


(jx  {-2      U      2 


.-.  7.r--3-  3.r  !  1,  and  .r  -  1. 


^     ^1       2.r-f84-       13.?:    -2    .x      7x      .r-f-lG 
y  17a;  -32^3      12         3G 

In  tliis  case,  taking  together  all  the  fractions  liaving 
only  numerical  denominators,  we  get 

Sx  f  34  +  12a;  -  21  x  +  .r  +  IG  ^    13>r-2  . 
36  "^  17a;- 32' 

25       13.r-2 


or  —  ^ 


18      17a; -32 
hence,  a;  —  4. 


.•.425a;   -800  =  234a; -3G 


5. 


6. 


It  is  often  advantageous  to  complcfc  the  divisions  repre- 
sented by  the  fractions. 


.    Solve -3 —  X f  1 


33 


X' 


54 


Here,  'completing  the  divisions,  we  have 

4a;_17_l  ,2x^  ,  _G     x 
9        9       9"^  3"      ^~   a;      9' 

10a;      o  ,  a;      G 

-^ 2  =  a;  + :: 

9  9      a; 


Therefore,  -2 


-,  or  a;  =  3. 

X 


7. 


3  L.C.M.  of 
raction  willi 


uul  .r  —  1 


IG 


ions  having 


o 

312*' 


-30; 


\8xons  repre- 


LINKAH    KurATlONS. 


1  •"^, 
1  i  <J 


.     <ix  -f-  h  .  ex  'Y  d  ,  ^ 

x  —  m       X  —  n 

,  am  -\-b  ,       ,  cn-\-  d  , 

.'.  a h  e  -\ —  n-^-r. 

X  -  n\  X  -   n 

:.(am-\-h)[^v.      n)-\'{f'n'[-d)(x-    tfi)      0. 

.*.  ((fni  -|-  />  -h  en  h  ^0-'^*  "  (''  +  '*)  ^"''  "i"  ^•'^^  'I  '^^'^• 

5.    Similarly  may  l)o  solved 

x  —  tn      X  -  n       (.r  —  ?/? )  (x-  -   //) 

.  ahi-\-h      en-]d[c{in-{-')i)-\-j\x~einn  —  y__ 
X  —  in       X  -  -  n  {x  -—  m)  (x  --  ri) 

.".  {(im  4-  ^>j  (.^'  —  w)  +  (cm  -h  (^)  (a*  —  ///) 

+  [c (//«.  -}-  ?i)  -\-f'\x  —  6'?«n  —  y  -    'X 
.-.  [(,,  _^ ,.),,,  _}_  /,  .|-  (,,  .|_ ,.),,  -1- ,/  _|.y  ]^,. 


6. 


132.r+l   ,  8.7;  + 5,    .^ 

3.T4-1  07-1 

43 


.-.44 


3:1  +  1 


4- 8 +  -^-52,  or     ^'^ 


43 


X      I 


X 


1      3.1  +  1 


39.1-  +  13  =  43:c  -  43,  ami  x  =  14. 


25 


^ 


a:  ,  162:  +  41__ 


:c+l 


+ 


3;r  +  2 


■5-5  + 


23 


+  1 


Taking  the  last  fraction  with  the  first,  ami  multiplying 
the  resulting  equation  Ly  15,  we  liave 

240a:  +  G3_.7.  ,  5  a; -30 


2>x-\- 


=  75  + 


x-\-l 


80 


97 


or 


3.T  +  2 
97  35 

3a;+2      x+l' 


75  +  5  - 


8. 


35 

x-\-l' 

.. .  -  07 


ana  x 


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170 


LINEAR   EQUATIONS. 


8. 


X       ft    I    X       0    I    X        C Q 

h  -\-  c      a-\-  c      b  -{-  c 
. x—a      1 , x~h 


1  +  - 


1  + 


X 


-1-0. 


h  -\-  c  a-\-  c  b-{-a 

'  ^  —  Qj' -\- ^ -{- c)  I  X  —  (a -j- b -\- c)  .  a;— (a+^  +  g),_n 
b  -\-  c  a-{-c  b-{'a 

which  is  satisfied  hy  x  —  (a-\-  b-^  c)  =  0) 
\  x  =  a-}-b-\-c. 


9. 


7n 


x  —  a 


+ 


n     _'m-\-n 


x  —  b 


X  —  c 


mix  —  c)  ,  n(x  —  c)  , 

x  —  b 


x  —  a 


m 


I^Zl£)  +  !^i^^O.     (See  Exam.  4.) 


10. 


x  —  a  X 

.'.  [m(a  —  c)-\-n(b  —  c)]x  —  'mb(a  — c)  -{-na(b  —  c). 

l/i^-2\  .  l/v-l\     2rv-l\      5 


^.,H"^ 


18 


.  1  ,  ^_  ,  1  ,  __1 2  2     _  5 

'■3'^2/-5"^6"^y-7"  9     y-10      18 


+ 


y-5     2/-7     y-10 

y  --5      y  -  7 


=  0. 


-5 


4- 


y_5    '    y-_7 

.-. -82/  +  50-:0. 


0. 


2/ =  61. 


1. 


Ex.  51. 

10a: +  17       12a; +  2      50;  -4 


18 


13a; -16 


9 


Ga;+13      9a;+15  .  o_2a;  +  15 
15  5a;-25~^  6 


LINEAR    EQUATIONS. 


177 


\±t£}~o 
fa 


1) 

na(b  —  c). 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


10. 


11. 


12. 


13. 


X 


1        d\x-i-2, 


+  3f 


4a;- 7  .  2 -14a:  .  Sj+x 


2x 


9 

2x-{-a 


+ 


7 
3  a: 


14 


a 


3(a:-a)      2(a:  +  «) 
a: -4    ,  8a; -13 


-2i. 


6a;  +  5      18a;-6      3 
3a,' +1        a;  -11 


2a; -15 
x-9  , 

•T 


X 


2a; -10 
-5 


1. 


=  2. 


a;  —  5     x  —  8 
12  .    a; -4 


X 


+ 


—  9 


a;— 7       a:  — 12 


X 


7 


3a; 


X  -  13 
a;-2 


19  ,  3a: 


11 


+ 


a;+7 
a;-l 


=  6. 


10 -34  a: 
2 


2a;+l      3(a;-3)      6 

a;+l      ,     a:  +  4    __  9 
4(a,-+2)"^5a;+13     20' 

5(2^>-  +  3)      5-7_^-^^^,_e. 
2a;+l         2a;-5 


19 
21' 


14. 


15. 


16. 


X 


-7 


+ 


a: 


9      x-S 


7  a: +  55      3a;__o      3.r"'  +  8 


2a; +  5 


-ii:^=9 


17 


+ 


15 


',x 


32 


a;-16     a:-18     a;-17 


178 


LINI-LVR    EQCATIONS. 


17. 


18. 


1-253;       S~-2^x 


28 -5a;      10.r      11   ,  :r 


15 


14(.r--l) 


30 


+  ^- 


X 


2  +  2lx'-^x'  ^ 
6-  bx-j-x"^ 


ia; 


x  —  b 


19. 


80  +  Ga;  ,  GO  +  Sj 


x+l 


+ 


48 


.r|-3        x-\-l 


+  14. 


20. 


21. 


5 a:' +  3;  -3  _^  1x'      3x~-d 
bx-4:  7.^•-10 

.-r  —  9    _  a;  -f- 1  ,  x  — 


.r 


a; 


2 


+ 


8 


o;- 


o; 


1      a;-6 


^,2 


22. 


3  a; 


X 


\).x 


6x 


-'    -1x-17^2(x' 

x—0       ~  x-8 


1^, 


23. 


i£±l  +  i£±0 ^  4.y  +  6  ,  4.r+10 

4a;  +  5      4a;+7      4a;  +  4       4a;+8' 


24. 


25. 


\x 


3 


I X 


4      2a; -7      2 


a; 


8 


2a; -4      2a; -5      2a; -8      2a; -9 

7a.- +6  _  2a;-f-4|  ,  £__  11^ __ a;-  3 
28  23  a; -6      4"     21  42 


26. 


27. 


X        O    I    X 


11 


a;  —  6      a;  —  12 

2  5        o 


x—7.    x—d 
a;  -  8      a;  -  10 


X 


lU      2~Qx__^      5a;--K10-3a;) 


13 


—  X 


39 


28.    - 


29. 


10-17. 


+ 


1  +  a; 


+ 


27(.f'^-a:+l)      2(a;'^4-l)      54(.r+l)      d(x'+l) 
2a;'+.r-30  .  a;'+4a;-4„  a;^-17  ,  2a:'+7.r-13 


X 


~l 


-4 


+ 


^x — o 


30. 


x  —  a 
X  —  b 


+ 


X  —  h 


{a  -  hf       _  2{a-x) 


a 


(x  —  a)  (x  —  h)         a  -\- 


X 


11    ,    X 


m. 


|9  {x"  +  1) 

r-13 


LINKAR    KQTATinNS. 


170 


31.    12.M,1IO<^,-H.-MlTa_^^3^ 


'6x-\  a 


2.6-  + Da 


13|^.r-5      13|x'-1113.^.7;-7  ,    V6}^x 


13i 


33. 


+ 


1 


13^. 

X 


,    V^x  - 

9 

"^134.^- 

10 

16.7; 

2(:r-lV      %i:\)      2{x' -^  \)      {x    -l){x''  ^V) 


34.    i(fr  +  4)-iY^.--|(5    -1^ 


35.    ^ 


36.    1  + 


81.^2-0 


(3a;-l)(.f  +  3) 


2v  ^'  +  3  y 


57-3.7 


2x  {-I 


437  +  5  __  x''-~x-\-2  _  ^ 


37. 


2(a:-l)      2(a:-fl)      a:'' -2:^  +  1 
7.2; --30     5.r-7      2   -21:r 


lOi 


i^-3 


21 


42^;-  171  _  -^Q  ^  J-^^-;?    -1(4      7:r). 


63 


63  -  14:r 


38.    m^^^Qx  +  ^±^^\Z^      101-540; 


13-2 


X 


8 


8 


nr\     ^  —  y  .2;      o  —~  i.jjX      Q 
\-Zx      7-42' 


24  .r^  -5 


7-252'  +  12a;^ 


40     8^  +  25      16.7;+93__,  18.r  +  86      6.7; +  26 

V-^  '^  1-11  <^  I/-V  I 


41. 


2.r  +  5        2.r  +  ll 
1  1 


2.1-  +  9 
1 


V 


1x  +  7 
1 


x-\-  a-\-h      X  —  a-\-  b      x-\-a~b      x  —  a~b 


0. 


§  40.    The  results  deduced  in  the  sections  on  ratios  may 
often  be  applied  with  advantage. 


180 


LINEAR    EQI-ATIONS. 


1. 


2. 


,  7  Examples. 

ax_-j-_b_m 

ex  -{-  d      n 

.  (ax~{-  b)d~(cx  -\-  d)b  _  md  —  nb 
{ex '{-  d)a  —  {ax  -\-b)c 

vid  —  nb 


na 


mc 


'.X 


na  —  VIC 


a 
m 


4. 


ax^  -{-bx  -^  c 
inx^  -\-nx-\-p 

{ox"  -\'bx-\-  c)  —  ay? 
{nix^  -f-  ^^  + 1>)  ~~  '^'^'^^ 
,    bx-[-  c       a      , 
nx-\-p      VI 


a 

VI 


3.    - 


(Page  156) 


(Page  155) 


3.r+7__3a:-13 

:r +  4  x  —  ^ 

By  (5),  page  155,  each  of  these  fractions 
difference  of  numerators 


difference  of  denominators 

5 


.  2Q__3a:+7_3 

"  8'       a;  +  4  x-\-^' 

inx  -f-  g  +  ^ ??ia;  ■\-  a-\-c 

7ix  —  c  —  d      nx~b  —  d 

vix  -}-  a-\-b nx  —  a  ~  d 


1 

or  -  = 


2      a:  +  4' 


X  —  6. 


or  by  (2),  page  155, 


vix -\-a-\-  c      nx  —  b~d' 

vix  -\-  a-\-b  _7ix  —  c  ~~  d ^ 
b  —  c  b  —  c 

or  {n  —  vi) x=-a-\-b-{-c-\-d\  .'.  x  = 


5. 


+ 


+ 


x~6a     x-{-3a     x  —  2a     x  —  a 


LINEAR   EQUATIONS. 


181 


Transposing, 


'    +    2 


6 


X  —  Qa     X  -{-Sa     x  —  a     x  ~2a 
Sx-da  3a;-9a  0 


x'-Sax-lSa'     x:'~Sax  +  2d'     20a' 
.'.  3.r  —  9a  =  0. 

,' .  X  -^-  O ct. 


-by  (5)  of  §37. 


1. 


2. 


3. 


4. 


1-f.f 


X 


1 

a 


X  4~a 

— ■ —  —  m. 

X  —  a 

ax-\-  b  _  VI 
ax  —  b 

a-{-x 


n 


1. 


b-\-2x 
5.    '^^±^^~b. 


Ex.  62. 
8. 

9. 

10. 

11. 


x  -\-  m  _a  -\-  b 


a 


a 


6. 


7. 


a  —  x 
a 


a —X      b  —  X 

a-\-x     a-\-b 


12. 


13. 


14. 


X  —  VI 

a-[-b 

\-\-cx       \  —  CX 

a-\-bx      c-{-  dx 
a-^-b        G-\-  (i 

a4-  bx      c  A-  dx 

! :z=r  ! .  I 

a  —  b        c  ~  d 

a  —  X  _a-\-x 
b  —  X      b  ■[-  X 


2a;''-7a;+3 
ax-\-  b 


c 


16. 


17. 


18. 


a  —  X 

ir>. 


^x 


a 


a' 


a'x 


ax  —  b-{-o 
2c 


x'-^%x^2 


{b  +  oy 


a?-\-ab-{-  b^      a?  -  b^ 

7__  x-\~1 


a~b 


2  ex. 


2x 


3      a;+ll 
4a: -5       10a;  -  32 


2a:+i0       5a;  — 8J 

57a;-43_  39a;-7 
19a: +13      13a: +25" 


19. 


20. 


21. 


23a: +  5^  _    36a:     7 
115a: -29      180a:+23' 

210.r-73_21a:+7.3 


310a7 


31a:  +  8 

_mx—a~o 
mx~c—d      nx—b—d 


mx- 


■66 

ah 


182 


T-INEAR   EarATlONS. 


22. 


23. 


24. 


25. 


26. 


27. 


28. 


29. 


x^  +  CLX'  —  bx  -f-  c  _  .r  -r 


(/j;  —  /> 


ax' 


-\-  bx  -\-  c      31?  —  ax 


8.?;'  1   12:6-' --8a; +  5  _  4a-^ -f- G.r  -  4 
8.r''  -  12a:'  +  8a;  -f-  5  '  ^  4,?:'  -  G;/;  +  4* 


^  +7i+--iOH  '^>  +  ^'y- 


aa; 


bx 


ex 


l/_a_ 
2V;ra; 


+ 


acx 


abxj 


1  —  ax  ,1    -  5a;  ,   1 


ac 


14? 


-  ca;  _  /2  J  2      2\  ^ 
ab         \a      b      c) 


+ 


b 

ac 


abc 


+ 


O-^i)- 


aa: 


{a  +  i)'      («  -  bf      (a'  -  ^>')  (a  -^b)      {a-  If 
X  -\-  a  x  —  h  ,  x-{-  c 


+ 


30. 


31. 


32. 


33. 


34. 


{a~b){c~a)      (a-Z>)(5    -.•)      {b-c)(c-a) 

ci  +  g 

{a  —  b)(b  —  c)  (c  —  a) 

/a;  +  2aY  ,        /a+2a;Y     ^ 

a:   — ! 4-^1  — ^ —  2  a. 

\x  —  a  J  \a  —  X  J 

x-\~a 


X—  a 


a' 


a;'  4-  «a;  4-  «^      x^  —  aa;  4"  «■*      ^  (^*  +  Q^ V  4"  o^*) 

a;4-«  _  /^^av-foj^Y 
a;4-6~\2a;4-6  4-(?/ 


+ 


1 


-J 


(a'4-a)-^'      {x^bj-a"^     x^-{a-\-by  '  :c»_(a-^»)' 

^^  /6a;4-45      7a;4-67\  ,  130-39/8^+5?  ,  9a;4-68\ 
V  a;4-l  a;4-4  j  \  ^+2         a;4-3  ^ 


wx     aoxj 


V. 


ax 


[a  -  ly 

c 

(c-a) 


aO 


9a;+68\ 
x+3  / 


MNKAll    EurATIONS. 


l.s;i 


§  41.    Sometimt's  a  factor  independent  of  x  can  be  dis- 
covered and  rejected. 


Examples. 


,      ?tahc      hx  ,      a 
1.    — --  -  -7  +  7-- 


2ji 


-  -  3  ex 


a  -f-  ^       a       {a  -j-  ^)'' 
Transpose  —  and  factor  ;  then, 


Mr  r2a±  h\ 


a 


_«A_r3H-  -^V-  ^hc+^i(\  -  ""^+^'^1 


=  X 


a^ 


2. 


*a  +  6 


a:-f-rt 


—  X. 


x  —  h 


x—c 


(a  —  b)  (c  —  a)      (ci  —  h)  {b  —  c)      {h  —  c)  {c  —  a) 

h  -t-_£ 

~  {a-h){b-c){c-a) 

Add,  term  by  term,  the  identity  (Th.  III.,  page  67), 


(a  —  h){c  —  a)      {a  ~h){b  —  c)      {b  —  c)(c  —  a) 

.  2x  h  +  c 

*  '  (r  —  b)  (c  —  a)      (a  —  b)  (b  —  c)(c  —  a) 


IM 


MNF:MI    KliUATlONS. 


3.    (x  -f-  a  -f-  bf  +  («  +  bf  -  {x  +  by  -  {x  4-  af 


-{•.rM-«"  +  ^' 


a 


he. 


The  Ic'ft-liand  member  vanishes  for  x  ~  0,  and  hence, 
by  symmetry,  for  a~0  and  b-O;  therefore,  it  is  of 


tlie  f 


(lb. 


orm  VKinx,  in  w 


hid 


1  1)1  IS  nimid'ic 


al. 


Put  X    -  ^«  --  A,  and  w  is  found  to  be  G. 
TTenec,  the  equation  reduces  to 


(S<,1 


)X 


th 


itoc,  ant 


I 


X 


\ 


(x  -  J 


.r 


-  2a  fi 


^v  —  b)      X  —  2  />  -f  a 

Let  .j;  —  A  ~  ni,  x  —  a  =  n,  and  hence,  7)i~7i~  a  —  b  ] 
then  we  have 

n*  __  n      (7)1  —  7i)      2n  —  wi 
7^1*      m  -f-  (9?i  —  7?)      2ni--n 


Bii 


'.  2?^i?i" 


?t*=  2m"/i 


?>t*. 


•.  7/t*  —  n*  —  2m'/i(??i^  —  n'^)  -  -  0. 

•.  (7/i^  —  ?l')  (77i^  +  ?i'''  —  27)171)  =  0. 

•.  ())i  +  ?i)  (?;i  —  7iy  =r-  0. 

it  7)1  ~  n  ==  a  —  b,  and  rejecting  this  factor,  which 
does  not  contain  x, 

7n-\~n  =  0. 

But  7)1  -}-  71  =  2x  ~  a  ~  b. 

.\^,x  —  a~b  =  0. 

.'.x  =  ^(a-\-b). 

Ex.  53. 

1 .  a(b  —  x)  -{-  b  (c  —  a;)  =-b{a  —  x)  -\-  ex. 

2.  (a  +  Z>a;)  (a  —  5)  —  (aa:  —  5)  =  a6  (.r  +  1). 

3.  {a  —  b){x~c)-\-{a-\-b){x-\-c)  =  2 (bx  +  ad), 

4.  (a  -  Z>)Of  -  c)  -  (a  +  5) (:r  4-  c)  +  2a(6  +  e)  =  0. 


LINEAR    EQUATIONS. 


185 


I,  and  hence, 
sfore,  it  is  of 


?i  —  a  —  6  ; 


factor,  which 


la 


6.  (a~lj){a    -c){^n   \  j-)   \  i^n -\  fj){(i  \- c){a  -  x)      0. 

6.  {a  —  h)  (a  --  c  -{-  x)  -\~  (a  -\-  b) ((t  -{-  n  —  x)  —  2«t\ 

Solvo  in  (x  —■  c). 

7.  (?/i  +  ^0  i'f'  l  f*  -  •'■)  +  ("      '^^0  {^^  ~'  '^)    '  ^'  (^^'  +  ^0- 

8.  in (d  -^  b  —  x)  —  71  {k -    a  —  b). 

9.  {ni  -\-  7i)  (m  —  n  —  x)  -f-  m  {x  -  n) -—  n  (x     in)  —  7n^ 

10.  !^^z::?4-!Lzi:-|iillf ::3. 


?r 


7/i  71 

(I'^b  -  -  X        It^C 

X  J.  •       —  ■■ —  — |—  

a  ( 


V 

X   .   ckt  —  X 


-h 


:-0. 


12.    ^11^  +  ^- 
be  ca 


3'         C         X f\ 

—  -] ;—  —  U. 


ab 


-  „     I  —  nx  .  1  —  bx  ,   1  ~  ex      <-. 
Id.    ~ 1 1 —      u. 

he  ca  ah 

Deduce  the  solution  from  that  of  No.  12. 

- .     a  —  bx.b  —  cx.c  —  ax      Ci 
he  ea  ah 

15.   («  +  J  +  .) X - 2l+4'  =  2"if  +  h+J>\  0. 

a~  b       a -}  h      \a  —  hj 

3 abc  ,       d'b''      ,  (2a-\~b)i;'x  __  (/>  f  3 a£)x_^ 


16. 


a-f-Z;      (a-|-/*y        a{a  +  by 


a 


17.    --f 


10  .  4      9 


o 


a; 


9 


-  +  ~ 


Solv 


e  in  — 


7,1      23 


18.    -  +  -  = 


X 


X 


~  + 


12     4. 


19. 


20. 


7  ,  13_2(5a,--12)      17  ,10 


3  "^50: 


20 


+ 


10-a;  .   13  +  a;_7:i-  +  2G6      4a'+17 


+ 


:r  +  21 


21 


18G 


LINEAR    EQUATIONS. 


21. 
22. 


+ 


8 


1 


x  +  S     2(a;-f3)      2      2(xi-S) 


X 


8a; -15 


11 


3 


23. 


n  — 

X 


(I 


+  7. 


1 

X 


X  ~ 


(I. 


24. 


a 


(I 


b  -  -'— . 


X 


a 


d-^- 


X 


25. 
26. 
27. 

28. 
29. 
32. 
33. 
34. 
35. 
36. 
37. 

38. 
39. 

40. 
41. 
42. 


{x  -  1)  (a:  -  2)  -  (a;  -  3)  (a:  -  4)  -  3. 
{x  ~  3)(a;  -  4)  -  {x  -  2)  (a;  -  G). 

2(.r  -  4)(3.f  4  4)  -1-  (2.^  -  3)(3ar  -}-  2) 
-6(a:-2)(2.r-  3)^.0. 

(a   -  x)  {h  -  x)  ^--  x\         30.  {it    -  x)  {h  +  x)  :=!?-  x'. 

{a  -  x)  (x  -  h)  =  x'  -  6•^    31 .  (x  ~  a)  (x  -  b)  ■-=  x'  -  a\ 

{a  -f-  x)  (b  -h  ^)  =  {(t~  x)  {b  —  x). 

(ax  -f-  b)  (bx  -\-a)  =  (b  —  ax)  (a  —  bx). 

(a  —  x)(b  —  x)-{-  (a  —  c  —  x)(x  —  b-^c)-=  0. 

(a  —  x)  {b  —  x)  -{c~x)  (d  —  x)  =-  (c  +  d)  x  —  cd. 

(x  -  a)  (x  -h)-  {x  -c)(x--d)  -.  (d  -  a) (d  -  b). 

[(a''~b')x -ab][a  -  (a  +  b)x]  +  2ab''x 
=  [(a  +  byx  +  ab][b  -  (a~b)x]. 

(.r  4- l)(a:  + 2)(:i'  + 3)  =  (a:  -  3)(.i'  + 4)(a;+ 5). 

(a:+l)(a:  +  2)(a;  +  3) 

-(a;-  l)(a;-2)(a:-3)  +  3Cr+l)(4.r-f  1). 

(:.•  +  1)  (^  +  4)  (a:  +  7)  -  (ar  +  2)  (a; -1- 5)^ 

(a:  +  2)  (a;  +  5)^  =  (a;  +  3/ (a;  +  0). 

(x  -    1)  (x  -  4)  Or  -  6)  -  x(x  -  2)  (a-  -  9)  -  13G. 


LINEAR   EQUATIONS. 


•ir 


".  ■» 


X 


V)  :=  h' 


-'i 


X 
i      ..1 


0. 
\)x  —  cd. 
\){d--h). 


+  5). 


+  !)• 


13G. 


43.    {a-\-x){Jj-tx){c+x)-  {a-x){h-x){c-    x) 


44. 


{x -it) {x  -h){x-c)-{d-a){d-   h) (d-  - c) 


X 


-d 


— ^={x  -dy 


46.    x{x  -  ay    ^  {x  -  a  h  /-*) C'f  -a  +  r) (x      h  --  r) 

46.  (u,--(«-f^*)(u:--i-f  c)(.r-c-f  fZ)    -a.-*(a.'     a -}-(/) 

47.  {x--a-\-h){x~  h-^c){x  —  c-[-d) 

—  x(x  —  a  -}-  c)  (x  —  c  -{-  d)  ~  br(d  —  a). 

48.  (^'  -2a)  (x-2b)  (x~-2c)  -  (x-a-b)  {x    h-c)  {x  -c-a) 

=  (a-ib+c)  {a'+b'-^-c')     dabc. 

49.  x^--(x  —  a-}-b)(x  —  b-}-c){x~c-\-ci) 

=  (a+b-j-c)  (a'  4-  b" +c')~2  (a'b  +  b'c + c'a)  -  ^abc. 

51.  (:r  +  ci)(x^b)-{-{x  +  c)(a;  +  «)  =  (x-i-b){x  -\-  d) 

^{x  +  d){x-\-c). 

52.  {ax  +  />)  (aa:  —  c)  —  a  (6  —  a:)  {ax  -f-  i) 

=  rt^  {x  -  6')  (a;  —  b)~-a  {ax  —  c)  (c  -  -  x). 

2x-^  ,  3.r"2      5a;'^-29a7-4 


53. 


54. 


55. 


56. 


57. 


x-^        x~S       x'~-  12ar  -f  32 
5a:-l         3a:  +  2  __  a:' -- 30  a;  +  2 


3(a;+l)      2(a;-l)  6a;^-6 

3a:  -7      3(a:+l)^       lla:  +  3 


2a; -9      2(a:  +  3)      2:1-'' -3  a,- -27 


'x  —  b  ,  8a;-7 


10a:+7 


ia: 


2      3a;-l      9:i-^-9a;  +  2 


-5. 


2a:+7      3a;-6  .  5(a;-l)_  3ar-2  ,    bx-%       2ar4-2 


+ 


+ 


+ 


dX — (         ^X — D  u  X     ^0         ^X      0         uX — «jO        dX~~l 


188 


LINEAR   EQUATIONS. 


xi 


58. 


59. 


Ax 


1-i-x 


X  —  a 


X 


x^-l' 


X 


X  —  VI     X  —  n 


-0. 


60.    . 


1 

1  .    -^4     1 


X 


i+- 


2x 


1 
4 


61.   ^  + 

c      ax 


ex 


ax 


b     a     cx  —  h 


62. 


3 

2 


1 

x 


2 
3 


1 

x 


32 

2     3 


3+1 


2 
3 


I        1 


X 


'Ix 


fl 


g3^    2(.r  -1)  J  :r  +  8^3(5a:+16) 
:i'  -   7        a;  — 4         5a:  —  28 


64. 


65. 


66. 


ax 


+ 


ex 


a      c 

—  zr  +  z' 


nix  —  p      nx  —  q.Tn      n 

ax-\-h      cx-\-  d      a      c 

= — 

mx—p      nx  —  q      m      7i 

h  —  X  ,c  —  X _a{j—  2x) 
a-\-  X     a  —  x        a^  —  x^ 


67. 


68. 


a-\-h      h-^c     a  +  c  +  2Z> 


x~a 


x  —  b' 
bx 


x~c 
ax 


ax^jzJ^  _     bx     ax     __  (ax"^  —  2b)b 

ax  —  b      ax  -\-  b      ax  ~b         d^x'^  —  b'^ 


ft.- 


69. 


ax  —  b       ex  -—  d      (bn  +  dm) x—(bq-\- dp) <*  j_ ^ 

mx-'p      nx -~  q  (7)ix~p)(nx — q)  in      d 


LINEAR   EQUATIONS. 


189 


70.  ^?_-f-!L_+_Zi_^_^?L_+_ii_+    P 


x  —  a      x  —  b      r  ~  c      x  —  c     x  — 


a 


x  —  b 


71. 


72. 


ax  —  2a  _ax  —  2b 
ax  — 2b      ax-\-2a 

1      1  1 

a 

a      X  X 


75. 


-+-      a+- 
a      X  X 

ax^  —  bx^  -f  ax  —  d 


73. 


74. 


7.'c'^-4a:  +  2~7' 
ax^  —  bx-\-c 


a 


tna^  —  nx-{-p      m 


ax 


~b 


ma^  —  noc^  -\-  mx  —  q      thx    -  n 


76. 


1 

4         .  1 


X 


4  +  ^^ 


4      1 


1.     77. 


n 


X 


3     2      2 


l^  + 


T  +  ^r 


2 
-  —  A' 


3      3^ 


X  —  - 


78. 


21 


71 


21 


71 


a; —  98      .r-94      a; +  44     a;  — 52 


79. 


.r  —  6 


+ 


9 


X 


11      .^-7 


-I- 


X 


o 


80. 


9 


9 


2 


a: -51      .r  — 15      a; --81      a; +  81 


81. 


+  ~-^  ---= 


8 


x—G     x—9     x—7 


+ 


X 


10 


82. 


+ 


8 


=  ~-^  + 


a;  —  G      .T  —  3      a:  —  2 


a; 


-5 


83. 


on  —  n      a 


771  — n     a 


x  —  a      X  —  in 


X 


X—  n 


84. 


a 


-{■b      a-fc b-\-d c-]-d 

X    h      x-c~ x—{a-\-b-\-2c^d)      x    {a-{-%-lc--\-d) 


190 


LINEAR  EQUATIONS. 


85.    (x-{-a  +  by  —  {x-\-a)*-(x-]-by+x'-(a+by-i-a*-i-b* 
=  12  ab  [x'^  (a  +  by]. 

„-      a  —  x    ,    b  —  x    ,    c  —  x  Sx 

ob.     —  -\ 1 = • 

ci^  —  be     b'^  —  ca     c^  —  ab      ab-\-bc-\-  ca 

87.  (^  — ^)(^-<^0  I  (n-p){x—b)   ■  {p~m){x~c)^Q 

b-\-e  c-\-a  a-\-b 

88.  (x-\-a-]rby-{a-\-by-{x-\-by-(^x^ay-{-x'-\-o!'-{-b'' 

=  \0abx{2x-^a-\-b){x^a  +  b). 

ftQ       <^^—  1 I     bx  —  1    J ca:  —  1  3  a; 


d\c-^b)      b\c-]-a)      c'{a-\-b)      ab  +  bc  +  ca 

=  3. 

Sx 


on      ^  —  26t     ,     37  —  26     ,     x~  2c 
b-\-c  —  a     c-\-a-—b      a-\-b  — 


91       ^  —  2a     .     x  —  2b     I     x  —  2c 

b-{-  c  —  a      c  -\-a  —  b      a-\-  b  —  c     a-\-b-\-c 


„„      a  —  x    ,    6  —  .r         c  —  a; 


3 


d^  —  be      b'^ 


be      c^  —  ab      a-{-  b-\-c 


93      a:4-  2aZ>    .    2ft&  —  x    _  rr—  2a6    .    x -\-  2ab 
a-\-b  -{-  c      b  -\-  e  —  a       a  —  b-\-e     a-\-b  —  c 

a  ,  b 


94. 


+ 


a  —  c  .b  —  c 


x-\-b  —  e     X  -\-  a  —  c     x  -\-b     x-{-a 


_       tn^  {a  —  b)      n^  {b  —  c)     p^  {c  —  d) 


X  —  7?2 


X  —  n 


X  —p 


96. 


q'^[pd-{-{n—p)c-]~{in  —  n)b  —  7na'\__^ 
X  —  q 

{x-2)  {x-b)  (ar-6)  (a;-9)  +  («+2)  (a-4)  (a-5)  (a-1 1) 

X 

^  (6+l)(^+5)(H8)(H12)  ^  (^_4)  (^_7^ (^_ii) 


+ 


(«>»    l)(a-8)(a-10)+(^>4-2)(H3)(H10)(^>+ll) 


x 


LINEAR   EQUATIONS. 


191 


by-\-a'^h' 


x—c) 


0. 


c^+a'+i' 


X 


c-\-  ca 


-\-c 


2  ah 


Equations  Resolvable  into  Linear  Equations. 

§  42.  In  order  that  the  product  of  two  or  more  factors 
may  vanish,  it  is  necessary,  and  it  is  sufficient,  that  one  of 
the  factors  should  vanish.  Thus,  in  order  that  (x—a)(x—b) 
may  vanish,  either  x  —  a  must  vanish,  or  x  —  b  must  vanish, 
and  it  is  sufficient  that  one  of  them  should  do  so. 

Hence,  the  single  equation  (x  —  a)(x  —  b)  =0  is  really 
equivalent  to  the  two  disjunctive  equations, 

X  —  a  =  0  or  x  —  b  =  0, 

for  either  of  these  will  fulfil  the  conditions  of  the  given 
equation,  and  that  is  all  that  is  required. 

Similarly,  were  it  required  to  find  what  values  of  x  would 
make  the  product  {x—a)(x—b){x--c)  vanish,  they  would 
be  given  by 


X 


a 


0,  or  X  —  b  =:  0,  or  X  —  c  ==  0. 

.'.  x=^  a  or  b  or  c. 

Hence,  the  single  equation  (x  —  a) (x—b) (x—c) 
equivalent  to  the  three  disjunctive  equations   • 

X  —  a  =  0,  or  x  —  b  =^0,  or  x  —  c=^0. 


=  0   is 


=  0. 
-5)(a-ll) 

-7)  (^-11) 
-10)(H11) 


Examples. 

1.  Solve  a;'^  —  a:  -  20  =  0. 

The  expression  =  (x—  5)(x-\-4:),  which  will  vanish  if 
either  of  its  factors  does ;  that  is,  if  a?--  5  =  0,  or  a; -|- 4=0. 

.*.  X  =  5,  or  .r  =  —4. 

2.  Solve  X*  —  x^~  x"^  -^x  =  0. 

This  gives  a^  (x  —  1)  —  x(x  —  l)~x(x- 
=  x(x  —  l)(x+l)(x-l), 
which  vanishes  for  a:  =  0,  a;  =  1,  x  = 


-1. 


192 


linI':ar  equations. 


3.  Solve  x"  {a  -  h)  |-  a'  (b  -  x)  +  h'  (x  -  a)  =  0. 
.-.  x\a  -h)—x{a^-  h')  -^ab{a-b)^  0. 

.'.  (x  —  a)  (x  —  b)  {a  —  b)  =  0. 

If  a— i  =  0,  the  given  equation  will  hold  irrespective 
of  the  values  of  a:— a  and  x—b^  and  therefore  of  the 
values  of  x ;  but  \i  a—b  be  not  zero,  then  must  either 

a:  —  a  =  0,  or  a;  —  i  ~  0. 

.•.  X—  a,  ov  X  =  b. 

4.  Solve  221  ar^  -  5  a:  -  6  =  0. 

Here  we  have  the  factors  17a:  —  3  and  13  a:  -f-  2 ; 
hence,  the  equation  is  satisfied  by 

17a:  — 3  =  0,  ora:  =  -^, 

and  also  by  13  a:  -f-  2  =  0,  or  x- 


-_    2 


5 .    Solve  (a:  -  of  +  (a  -  bf  +  {b-  xf  =  0. 

The  expression  is  equal  to  3  (a:  —  a)  (a  —  b)  (b  —  x),  and 
therefore  vanishes  for  a;  —  a  =  0,  or  a:  =  a  ;  and  for 
x  —  b  =  0,  or  x  =  b. 

Ex.  54. 

1.  If  an  equation  in  x  have  the  factors  2a:— 4  and  2a:  — 6, 

find  the  corresponding  values  of  x. 

2.  If  an  equation   give  the  factors  2a:— 1   and   3a:  —  1, 

what  are  the  corresponding  values  of  x  ? 

3.  If  an  equation  give  the  factors  3a:^  — 12  and  4a:  — 5, 

find  the  corresponding  values  of  x. 

Find  the  values  of  x  for  which  the  following  expressions 
will  vanish  : 

4.  x'-,2x+l',   4a:'^-12a:  +  9. 

5.  9a:'^-4;    x^-{a  +  by 


^^  •  oc^  —  2ax-{-a^. 


LINEAR    EQUATIONS. 


193 


6.  a;'-9a;  +  20;    4a;' -  18a:  +  20. 

7.  x^-\-x-Q;   x'-x-U]    dx'  —  dx-2S. 

8.  Gx'~12x+e>;    6x^~lSx-\-Q>;  6x'-^^20x  +  6. 

9.  Qx'~~5x~6;    (jx' -  S7 x +  6  ;    6x'  +  x-12. 

10.    A  certain  equation  of  the  fourth  degree  gives  the  fac- 
tors x^—x  —  2  and  4  a;'*— 2 a;— 2.    Find  all  the  values 

of  37. 


Find  the  values  of  x  in  the  following  cases 

11.  x^  —  2bx^~Sb''x^^0. 

12.  x^  —  ax"^  —  o^x  -{-  a^  =  0. 

13.  ar'-3a;  +  2  =  0. 

14.  X*  -  -  2  aa:"  4-  2  aJ'x  -  a*  =  0. 

15.  x'-\-(b  +  c)x^-hcx  —  h'c-bc''-=0. 


16. 


X  —  a 


X 


+ 


X 


—  b      X  — 


{a  -  by        _ 


-  ir 


a 


(x  ~  a){x  —  b)      {x  —  a){x  —  b) 


17.  a^~bx^-a''x-\-a'b  =  0. 

18.  S:i^  +  babx^-4:a''b^x--^a'h'  =  0. 

19.  x^(a-b)  +  a\b-x)  +  b'{x  —  a)  =  0. 


20. 


(x  —  b)(x  —  c)  .  (x  —  c)(x  —  a) 

(a-  b)(a-  c)      (b  -  c)(b  -  a)~ 


21.    X 


ft 


-2aV 


+«; 


+  a 


ix  —  a 

x-\-a 


X' 


a' 


22.    {x-{-a-{-bf-o(^ 


a 


b'^(x  +  a)(a'-b'). 


23. 


a6 


+ 


bx 


+ 


aa: 


(b—a)  (x —  a)      (x—a)  (a—b)      (ct—b)  (b—x)      a—  b 


24.    Form  the  polynome  which  will  vanish  for  a;  =  5  or  —6 


or 


104 


MNKAll    KlilATlONS. 


26.    Form  Dw  jiolyiionni  wliirli  will  vaiiisli  ior  x   -a  or  4r/. 
or  3  a  or  -  4  a. 

26.    Form  ilic  oqmiiion  wlioso  roots  arc  0,  1,  -   2,  and  4. 

§  43.  Fiinployiiig  tlio  langnago  of  Algebra,  tlio  principlo 
illiistnited  in  tho  procodiug  wction  may  bfl  Rtatod  as  lol- 
lows: 

Definition.  Any  (juanlily  wliicli  Knl)stitutc{l  for  .r  mak(»s 
tho  (\rp)rssio)i/(A:)  vani.sh,  is  said  to  ])c  a  root  of  tho  ajua- 
tlo)if{x)  —  0.  Thus,  if  «  bo  a  root  of  the  equation /(.r)  —  0, 
thon/(a)-=0. 

By  Tb.  I.,  if  .r  — a  is  a  factor  of  the  ^?o/v/??o??7<" /(.?•)",  then 
/(a)"  =  0,  and  a  must  bo  a  root  of  ibo  equation  /(■^)*'--  0  ; 
lionco,  in  solving  tlio  equation,  wo  are  merely  finding  a 
value,  or  values,  of  x,  wliicli  will  riiake  tlie  corresponding 
polynome  vanish.  Suppose /(.i*)**  =  C^  —  ^)  *A ip^Y  '  "^  ^»  we 
are  required  to  find  a  value,  or  values,  of  x  wliich  will  make 
(.r  -a)^(a:)*'~^Wanisb.  The  polynome  will  certainly  vanish 
if  one  of  its  factors  vanishes,  whether  the  other  does  or  not, 
and  will  not  vanish  unless  at  least  07ic  of  its  factors  vanishes. 
Hence,  (.?' —  a)<^(.r)"~^  will  vanish  if  a;  — a  =  0,  quite  irre- 
spective of  the  value  of  ^(.r)"~\  Also,  if  <^  (a:)""^  =  0,  tho 
polynome  will  vanish,  irrespective  of  the  value  of  x—a. 
It  follows,  therefore,  that  if /(a;)**  can  be  resolved  into  two 
or  more  factors,  each  of  these  factors  equated  to  zero  will 
give  one  or  more  roots  of  the  equation /(.r)**  —  0. 

When  there  can  be  found  two  rr  more  values  of  x  wliich 
satisfy  the  conditions  of  given  equations,  they  are  some- 
times distinguished  thus  :  ari,  a^j,  x^,  etc.,  to  read  "one  value 
of  .r,"  "a  second  value  of  a:,"  "a  third  value  of  a:,"  etc. 
Thus,  if  ^^  __  ^^  ^^  _  ^^  ^^  -  c)  =  0, 

. .  X\  —  a,    X'^  —  t?,    a'3  —  c. 


MNKAR    KQUATIONS. 


I'jrj 


,  and  4. 
^  principle 

J  0(1    flH    lo]- 

for  X  niakoH 
'  t.lio  cqua- 

•/(r)",  ilien 
t/(.r)---0; 
V  finding  a 
r  responding 
;)"-  ^  rr=  0,  we 
h  will  make 
linly  vanish 
does  or  not, 
)rs  vanishes. 
,  quite  irre- 
"- 1  =  0,  the 
e  of  X-  a. 
d  into  two 
to  zero  will 

of  X  which 
are  some- 

"one  value 
of  a;,"  etc. 


EXAMPLKS. 

1.    Solve2r'-13.r'-|-27j;-18    -0. 

Factoring,  {x      2)  {x      3)  (2  a;  -  3)  -^  0. 


3. 


H. 


2.    .?;■'      (a  l-^»)a;-|-(a-}-c)/>»--(a-|  <7)r7. 
.-.  .r'      {a  +  />)a;  -|-  («  -\-  r)  (b      c)  -.   0. 
.'.  .^'      [(^  4-  c)-\~(b~  c)]  a;  +  (rt  +  <?)  (/>  -  c)  =  0. 
.-.  [x  -  (a  +  ^)][a;  -  (h  ~  c)]  ^  0. 


Xt 


a-f 


-/>- 


.i; 


X 


(a'-\-b')x--(d'  ~h') 

(a'-  h-')x~(a'-\-lj^) 

±l_(l(x--l} 


x-l      b'(x~\~l) 

Xi-\-\        « _ 


e~!)'- 


a 


0. 


X, 


-1      6 


T  =  0. 


Xi  = 


__a-\b 


a 


^a  "r  J-    i_  ^ n 


a7o  — 


a 


~b 


X, 


a-\-h 


(a  -  -r)'  -I-  (^  -  xf 


_34 

(a  -  xf  +  (a-x)ib-x)  +  {b-  xf      49" 
.  (a-xy-i-2(a-x)(b-x)-{-(b-xy  _   2(49) -34   __ 
•  •  (a-xy-2(a-x)(b-x)+{b-xy      3(34)-2(49) 


16. 


[: 


(a  -x)  +  (b 


(a  —  a;)  —  (i 
(a-Xi)-\-(b  —  x,)  _  ^  ^ 


4^-0. 


0. 


a 


Xi  —■  i(5J  —  3a). 


a  —  o 

(a;  —  a)(x  —  b)  ,  (x  —  b)(x~  ci)  _■, 
(c  —  a){c  —  b)      (a  —  b){a  —  c) 


VM\ 


MNF.AU    KgrATIONM. 


SmUIimcI  Ivnn  l>y  \ovu\  from  iho  i(l(MiHiy  (h(h>  pM^"»  (i?) 

.'.  (.r      r)(.r      a)  ~- 0.     .\j\      r,    .v.,      a. 

(>.    Kiiul  lluMVf//()//ff/ roots  of. r^      l'J.r'  }  f)!./:'     '.)0.r  |  J)()      (I. 

Knotoring  llio  lori-luind  iiumuImt  l)y  ilio  iiu'lliod  of  §  liT, 

(.r      2H.'-      '!)(•'•'      <>•'•  I    ')      <^- 
.-..r,       2,    .r,   -    i,    or.r^      ().r  |   7      0. 

Sinco  .r' —  G.r  [  7  cannot,  ho  ro.^olvrd  into  rationul  lac- 
t(>rs,  W(^  know  thai  it.  will  not  givo  r.'itional  roots ; 
thcn^t'or(\  .r,  2,  x^-~-\  aro  tho  only  valncs  thai, 
iuoi^t  ilio  oon(litii)n  of  tlie  prohhnn. 

In  order  thai  two  oxitrcssions  having  a  common  factor 
may  he  o(|ual,  it  is  necessary  eiilier  thai  the  common  factor 
shouhi  vanish,  or  else  thai  tlie  prodnct  of  ilie  remaining 
factors  of  one  of  the  expressions  shonUl  W  ecpial  to  the 
prodnci  of  the  remaining  factors  of  tho  other  expression, 
and  it  is  sufUcient  if  one  of  those  conditions  he  fullillcd.  Jn 
symbols  this  is 

If  (.r      (T)/(.r)  -  -  {x  —  a)  <j>  (.r),    ..x'l-^a  orf{x) 


<AW- 


7.    .I'-j--  —  «H — 
X  a 

1      1 

.•.  .V  ~  a== 

a     X 


X  -  -  a 


X 


a 


•.  X    -  a  =  0,  or  ax  =  i. 


ax 


1 

a 


8.    {x  +  a-\-h){x-\-h-\-c)  =  {x  —  ^a-{-h)(2x~'^a-\-2h- 

x-\'a-\-h  _  2  a:  -~  3  g  +  2  ^  -  c  _  g-  —  4  ft  +  ^  -  c 

x  —  2>a-\-b 


c). 


X  -{-h  -{-  c 


2>a-\-  c 
Page  155,  (5). 


I.INKAIl    K(|l  ATIONM. 


l')7 


Oil  ul'  §  li7, 


Uioiiiil  fiic- 
onal  rc^oirt; 
^aliu'!^    thill/ 

imon  tact  or 

iimon  fiictor 

3  remaining 

iial  to  Um 

expression, 

uliillcd.    In 


I-  b  -^ 

ge  155,  (5). 


.•.  .r,  -    a  —  h. 

j(.r,       ;Wr  I  A)       Wa,   \   r,      /.  ;r,       Or/,       A    I    2r. 

5).    (r    L>)(.r    5){./-    n)(.r    1>)  |  0/ I  1^)(//     Ui//     '')0/     ^0 
■I  (2f-i)(2|5)(2f8)(r|  11^) 
=  .r(.r    4)^r    7)(.r     I J )  |  (v/ |  ])(//    l)(y    8)0/    lOj 

f-(2|l])(2|;V)(z|l())(2|   11). 

* 

Let  .r'      .7''       llr,  //      y'      O.y,  and  2'      2Ml'^z. 

.•.(./;'|18)(.r'|  30)  |  (/A  22)(/f  2())|  (2' 1   12)(2'1  40) 
=  .r'(y|  28)-|.(//    l())(y/|  8)  I  (2' I  22)(2'hm)). 

.•../:''^  I  •18.r'  I  540  I  y^     2//     440  I  z''-*  |  522' |  480 
--  :v' '  j  28 ./;'  \  //"     2 //'     80  |  2'  "M  52  2'  |  G60. 

.•.20.7.'     0,  .v'-n.i:-(\  ^1=^0,  x,^n. 

Ex.  56. 

Wliat,  can  yon  dedueo  I'rom  i\w.  lollowing  staiemcnts  ? 

1.  A  •  Ji-=0.  3.    (a  -  /*)^;      0. 

2.  tI  .  y>'  .  r;-  0.  4.    12:^7/  -0. 

5.  Wlial  is  ilici  ditlereuco  between  tlic  equation 

(:c-5y)(:r-4v/|  3)-  0 
and  the  simultaneous  equations 

X  _  5y  -^  0  and  j:  —  4;y  +  3  --^  0  ? 

Wliat  vahiec,  of  a;  will  satisfy  the  following  equations? 

6.  .r(.i--a)  =  0.  11.    a(.t'"'  — r/)=-0. 

7.  rt^-(:r  +  Z')--0.  12.    tex'-^Vx. 

8.  (a:  —  a)  (^>^  -  c)  =^  0.         13.    :t-'   |-  (a  -  -  a;)'  --  a'. 

9.  aa:''  =  3aa;.  14.    x^ -\- {a~  xj=^{fx-2xf. 
10.    ;r'  =  (a4-^>)a;.                   15.    (a-a;y+(a;-^»)'^  =  a■^+6^ 


198 


hlNKAli    Ktil'ATIONS. 


I 


16.  (a  -    x){j;      h)  -\-  ab  -----  0. 

17.  {a~  xf      {a  -  X)  {x  -  h^  -|-  {x  -  by  ■=  a'  -f  ab  +  b\ 

18.  f"  —  (a  —  b)  X  -   ab      J. 

19.  .r'      (tt  h  ^  f-^')'^"*  i   0*^^  +  ^^  +  ^'^)^"~  ^^<^~-^' 

If  X  must  })e  ])OHitive,  what  value,  or  values,  of  x  will 
satisfy  the  following  equations : 

20.  (a;-5)(.i-  h4):-0.  23.    S.r"  -  lOo; -f  3 --0. 


21.  x''\~29x-~^0^0. 

22.  :r'-l'^:c- 84^=0. 


24.  :c*- 13a;' -1-36-0. 

25.  .r'-2a;''  — 5:i;-f  6  =  0. 


Solve  the  following  equations : 

26.  (a  ~  xy  +  (x  -  by  =  (a  -  i)'^ 

27.  (a  -  a;)''  -  (a  -  a:)  (a;  -  i)  +  (a;  -  6)'  -=  (a  -  by. 

28.  a'(a-a:)"^^i'(i-a;)l     29.    «^(i^-a;)' =  6\a-a;y 

30.  (x  -  a)-''  +  (a  -  by  +  (6  -  a;)'  --  0. 

31.  (a;-l)''--a(a;'  -1). 


32.    ^--^,^- 


a 


X 


-b 


c  -\-  X 


33. 


a 


-{-b  ~  X      a  —  c-\-  X 


a~  c  —  X     a-{-  c 


34.  {x  —  a -\-b){x  —  a -^ c)  =  (a—  by  —  a;^ 

35.  {x-ay-b''-\'(a-\-b-x){b-^c-x)  =  0. 

36.  {a-\-b-\-c)x^~{2a  +  b-\-c)x-]-a—0. 


37. 


a 


-Vb 


a 


+  h 


X 


38.  {a-xy-\-{a~by={a-\-b-2xy. 

39.  a;  (a  +  A  —  a:)  +  (a  +  />  +  c)  c  =  0. 

40.  (n  — ^)  a:'  +  (^  ~  w)  a;  -[-  ??i  —  w  =  0. 


41. 


aa: 


--5a;-f  c 


7nx—nx-\-p     p 


42. 


aa;' 


—  ^>.t;  +  > 


a 


-b  +  c 


ma^  —  nx  -\-p      m —n-{-p 


ijNKAii  l•u^^A■nuNs. 


VM 


ah  ^-  b\ 

=  0. 

1,  of  X  will 

3  =  0. 

6-=0. 
[;+6  =  0. 


b\a-x)\ 


-\-  X 


-Vc 


a-6-fc 


43.  \,v'\a'     i?;»-2(a4-^)A'  =  (a  — .r)(/>[./;)     (a  |  u-'X^    ./). 

44.  (2a      />    -af  I  9(a-/>)' :.(a-|-^-.2a,f. 

45.  {'la\2c  —  xf  =^  (2b  +  :i-)('^^*      ^'  -\'  3c  -  2.r). 

46.  (3a -5/»4-a.-)(5a      3^   -0,-)      (7a      />      3x)^ 

47.  (3a-^   /^  h.r)(3a  |-6-:r)    -•  (5a -}- 3/>»  -  3a,-)'. 

48.  a  (a  —  b)     b  (a  -  -  c)  x  -{-  c(b  —  c)  or*  ^--  0. 

49.  (ab\-bc-\-ca)(x'-{-x-]~l)\-(a-by 

=  (2ac  +  b-") (.v'  +  a;  [- 1)  -|-  (a  -  c)"^a:. 

50.  (x  \-l)(x-{-S)(x-i)(x--7) 

+  (a.-  -  1)  (a:  -  3)  (x  -f  4)  (:r  -  h  7)  .-  90. 

51.  (x-l){x-\~3)(x~5)(x-i-d) 


-\-(x-\-l)(x  -  S)(x  +  b)(x   -9)  -h  18  -  0. 


52. 


+  L-.3 


a; 


^• 


53.    a;  +  - 


a 


4-6 


a 


a; 


a 


a 


+  b 


56. 


57. 


a  —  X 
x  —  b 


a—x 


+ 


a--  -  b      13 


6 


a  -a; 


5-f^ 


H 


a 


54.    0,' == 

0      a 


X 


a  .  X 


a~x 


mi 


m 
n 


n 

VI 


X 


58.    -  -|-  _  = (- 


X 


a 


n 


n 


55. 


60. 


61. 


62. 


63. 


a 


-{-X  ,  b  + 


+ 


a: 


59. 


6  +  a;      a-[-^ 

lx-]-af-lx-by~    2ab 
(a-xy-(x-by_    iab 


(a-xy  +  (x     by  _  5 
{a-x)(x-b)     ~2 


(a  —  x)  (x  —  b)         d^  —  b"^ 

{a  —  xy  +  (a~x)  (x  -  5)  +  (x    -  by  _^  49 
(a  -  xy  -(a-x)  (x  -b)  +  (x  -  by "    19' 

2d'  4-  a  (a  --■  x)  +  (a  -\-  xy  _  3 
2a"'  +  a(a  +  x)  +  {a-  xy      2 


I 


200 


l,INi:.\R    KQT'ATIONS. 


a* 


64.  (5    xy-h(2~xy    17. 

65.  {x~-ay-\^{a'~hy-\-(h-xy^ar' 

66.  (a  ~  xy  +  (x  -  hy  -  -  {a  ~-  by. 

67.  (x  +  ay  -  (a  +  by  -f-  {b     .r)'  -=  (.r  -f  a)  (.r  -f  />)  (a  +  b). 

68.  .r^  -~  (a:  -  /^)»  -  (a;  -  a  +  Z»)'  -  a»  ^{x-  ay  +  (a  -  by 

•\-b'-^{a-b)c\ 

69.  (.r  +  ay  -  (x  +  by  -  (x  -  by  --  (2  ay  +(x-  ay 

+  (a  +  ^'y  +  {a~  by  --=  (a'  ~  b')  c. 

70.  {x     a-{-by  ^{x-ay-{-{x-by  ~x'+a'^~{a-by^.b\ 


71. 


72. 


(a  -  -  a:)"^  +  (^  -  ^)' 


-\\{a--by 


(a  -  .x-)*  +  (a:  -  ^»)*       ^^^  ^ 


73.  2  (a  -  xy  ~  9  (a  -  ^X-^'  -  ^0  +  14 {a  -  .r)^^;  -  by 

-  9  (a  -  A')  (a;  -  by  +  2(.r  -  by  =  0. 

74.  4  (a  -  xy  -  17  (a  -  a;)^.^  -  ^)'  +  4 (a;  -  by  =  0. 

Find  the  rational  roots  of  the  following  equations : 

75.  a;*-12a;'  +  49a:'-78a;  +  40  =  0.     Letz-^ar^-6 

76.  a;*-6ar'  +  7a;^  +  6a;-8  =  0. 

77.  a;*-10a;='  +  35a:2-50a;  +  24--0. 

78.  32a;*-48a--'-10a;M-21a;+5  =  0. 

79.  a;3-6a;^  +  5a;  +  12  =  0. 


X. 


80,    lla;*+10a;'-40a:  =  176. 


81.    -- 


9 


X     x  —  a     X 


\a 


+ 


X  ~Sa     a;  —  4 


=  0. 


a 


LINEAR    EQUATIONS. 


liUl 


82. 


83. 


84. 


85. 


14 


4- 


14 


+ 


a;  f  20     a: -1-5     .i--4      a:  -55     a* -40     x-2b 

2a; 4- 5a  __ x-\-Sa  ,  _x x  —  a  _  x-{-b(i  ,  2j:-  5(/ 

X  X     a      x  —  ^la      X  -'du     a:- -4a      ^-   5a 


a:  +  4  .  x-\-2.x-{-i_x  +  S 


xi-2 


+ 


31 


X 


+ 


+  - 


X 


+--;-f 


1      a? -2     a:-  3 


X 


20 


-f- 


8 


X        X      '  x.        X  '   ^  mj        X '   '  ij 


4- 


20 


31 


H — ^-- 
5     ar    -0 


0. 


86.    x" 


(a-h/J 


X' 


2cx\ 
1  -\-  ex 


-~b\n 


I-  V  VI  I- 


ex 

f'X 


} 


87. 


88. 


89. 


4a:*  +  4a«  --33a- 


V 


2a; -fa 


ll.i'  +  28 


+ 


J(4ay  -S.t-'a-f-O.ra*      2a'). 


3jar' 


17.r-i  70     a,'^      14a,-f- 40 


8 


+ 


8 


X' 


ar*   -G.r  +  5      ar*  -  14:6- -f- 45   .  .r*      10a;  +  9 


90.    G  (a  -  xy  -  25  (a  -  xf  {x    -  h)  +  38  (a  -  a-)'  {x  -  by 
-  25(a  -  a-)(a-  -  Z,)^  4.  6(a;  -  by  =  0. 


CHAPTER  VI. 

Simultaneous  Linear  Equations. 


§  44.  There  are  three  general  methods  of  resolving 
simultaneous  linear  equations  :  first,  by  substitution  ; 
second,  by  comparison  ;  third,  by  elimination.  The  last 
is  often  subdivided  into  the  method  by  cross-multipliers, 
jind  the  method  by  arbitrary  multipliers. 

In  applying  the  elimitiation  method,  the  work  should  be 
done  with  detached  coefficients,  each  equation  should  be 
numbered,  and  a  register  of  the  operations  performed  should 
be  kept. 


Ex.    Resolve  u  +    v  +      x  + 

u  +  2v  +  4  ,r  + 
u-{-3v+  9x  + 
u  +  Av  +  IGx  + 


y+       2-    15, 

83/+    162-    57, 

27y+    812  =  179, 

64y  + 2562  =  453, 


u  +  5v  +  26x  +  125y  +  6252  =  975. 


Register. 


u 
1 
1 
1 
1 
1 


(■2)- 

-(1) 

(3)- 

-(2) 

(4)- 

-(3) 

i^)- 

-(4) 

(7)- 

-(6) 

(8)- 

-(7) 

C^)- 

-(«) 

V 

1 

2 
3 
4 
5 
1 
1 
1 
1 


X 

1 

4 

9 

16 

25 

3 

5 

7 

9 
o 

o 

o 


:ii)-ao) 

(12)-(11) 

(14)-(13) 

M^^^)    •    •    • 

H(13)- 60(16)] 

^I(IO)- [12(17) +  50(16)]}     . 
(6) -[3(18) +  7(17) +  15(16)] 
1)-  (19) +(18) +  (17) +  (16)] 


y 
1 

8 

27 

64 

125 

7 
19 
37 
61 
12 
18 
24 

6 

(] 


15 


57 


1  = 

16  = 

81  =  179 

256  =  453 

625  =  975 

15=   42 

65  =  122 

175  =  274 

369  =  522 

50=    80 

110  =  152 

194  =  248 

60=    72 

84=    96 

24=    24 

1=      1 

=     2 

=     3 

=     4 

=     5 


(1) 
(2) 
(3) 
(4) 
(5) 
(«) 
(7) 
(8) 

(^») 
(10) 

(11) 
(12) 

(13) 
(14) 
(15) 
(16) 

(17) 
(18) 
(19) 
(20) 


SIMULTANEOUS    LINEAR    EQUATIONS. 


203 


All  examination  of  the  Register  will  show  how  easy  it 
would  have  been  to  shorten  the  process;  thus,  (10)  is 
(7)  -(6),  which  is  (3)  +  (l)-2(2);  similarly,  (11)  is 
(4) +  (2)    -2(3);  therefore,  (13)  is  (4)4-3(2) -3(3)- (1), 


etc 


,f    resolving 

abstitution  ; 

The  last 

i-multipliers, 

:k  should  be 
in  should  be 
)rmed  should 


z 

1=    15 

0) 

16=   57 

(^) 

81  =  179 

(3) 
(4) 

m  =  453 

25  =  975 

(5) 

15=    42 

(^^) 

65  =  122 

(V) 

75  =  274 

(8) 

69  =  522 

(^») 

50=    80 

(10) 

10  =  152 

(U) 

94  =  248 

02) 

60=    72 

(13) 

84=    96 

(M) 

24=    24 

(15) 

1=     1 

(16) 

=     2 

(17) 

=     3 

(18) 

=     4 

(19) 

=     5 

(20) 

Ex. 

66. 

Ive  the  following  sy 

stem« 

of  equations : 

1.    2a;-f3?/  — 41, 

4.  h^'-\y  =  ^. 

3a; +  2?/ --39. 

Ja;-f7/  +  5-0 

2.    rx'c+7?/      17, 

5.  \y-^\x    1, 

Ix-hy-    9. 

\y-\^    1. 

3-  \^-\-\y-^. 

6.    1.5a;  — 2?/  — 1, 

3.r  —  4y-  4. 

2.5a;  -  3?/ =  G. 

8.  !  +  !=§ 

X      y      u 


10. 


7.    3.5a;  +  2-iy  =  13 +  4|a;- 3.52/, 
2\x  -f  0.8?/  =  22|  +  0.7a;  -  ^y. 

03 

y 


11.    17. 


X 


3, 


0.4 


X 


y 


6 


16a; -^^  =  2, 

y 


3  .  8 


X 


+  -  =  3, 


X 


y 

15     4 


12.    ^-f-  =  4i, 


3      ?/ 


X 


-  =  4. 


6 


+ 


10 

y 


02 


1^^2/7 

a;  ?/ 

0.8  .  3.6 


13. 


5^  ,03 

0.7"^  y 


6. 


X 


+  —  =  5. 
2/ 


10 


X 


+  -  =  31. 

y 


204 


SIMULTANEOUS    LINLAR   EQUATIONS. 


14.    |:r-i(y+l)=:l, 


15. 


16. 


7 


(^+l)+f(y-l)  =  9. 


x-\-2y      2x  +  y 


Sx-2      C- 


y 


X 


+  3y 


x~y 

1x  ~  13 
3y-5 


-8. 


17. 


^  +  2y  +  l_o 


18. 


19. 


\x 

\x 


y  +  1 
1+1 


30, 


a;  +  3y+13 
0.4a; +  0.5y- 2.5 
0.8a: +  0.1?/  + 0-6  _1 
5a;  +  3y-23         2' 

a:+l      7/+2„2(a:--y) 


X 


-3 


=  23/- 


ar. 


20. 


\x 


y  +  3      a;-2y  +  3_^ 


'57 


4y  +  3  J  4a;-2y-9_^^ 


21.  20(a;+l)  =  15(y+l)  =  12(a:+y). 

22.  (.r-2)  :(y+l):(a;  +  2/-3)::3  :4:5. 

23.  (a;-5):(y  +  9):(a;  +  y  +  4)::l:2:3. 


24. 


^  +  3_.y  +  8^ 
x+l     y  +  5' 


:a: 


3   _5a;-6 

2(y+l)      5y  +  7* 

25.  (a;-4)(y+7)  =  (a:-3)(2/  +  4), 
(.r  +  5)(2/-2)=:(a;  +  2)(y-l). 

26.  (ar-l)(5?/-3):r^3(3a;+l), 
(a:  -l)(4y  +  3)-3(7a;-l). 

t 

*?7.    (a;+l)(22/+l)  =  5.r+    93/ +  1, 
(.r  +  2)  (3?/  +  1)  =  9a;  +  13?/  +  2. 


=  2, 

=  5. 

13 
-2.5 

30, 

+  0.6_ 
-23 

1 
2 

5      2{x- 

-?/). 

5 

=  23/  —  a:. 


SIMULTANEOUS    LINEAR    EQUATIONS. 


205 


28.    (3a;-2)(53/  +  l)  =  (5a;-l)(y  +  2), 
(3a:-l)(2/  +  5)=:(a;  +  5)(7y-l). 


29. 

a: +  2/  — 37, 
y  +  z      25, 

z  +  a;  =  22. 

37. 

a;  +  y  +  z  —  3, 

2a;  +  4?/  +  8z-13, 

3a;  +  9y  +  272-34 

30. 

2.r  +  2y-7, 
7a:+9z-29, 
?/  +  8z-17. 

38. 

a;  +  2y  +  3z-32, 
2a;  +  3y  +  z-42, 
3a:  +  ?/  +  2z--40. 

31. 

1.3a; -1.93/-= 
1.7y      l.lz- 
2.9z-2.1a;--- 

1. 
2, 
3. 

39. 

a;  +  3/+2z  =  34, 
a;  +  22/  +  z-33, 
2a;  +  y  +  z  =  32. 

32. 

5a;+3y  +  2z- 
5a;  — 3y-39, 
3y-'2z-20. 

=  217, 

40. 

3a;  +  32/  +  z-17, 
3a;  +  y  +  3z  =  15, 
a;  +  32/  +  3z  -13. 

33. 


"J 


a; 

a; 


*2/ 


-a 


-iy==2. 


34.  Ha;+l|-y=10, 
2|a;  +  2|z=20, 
3i2/  +  3|z  =  30. 

35.  .r  +  3/  — z=:17, 
y  +  z  —  a;  -U3, 
z  +  a;  — y  =  7. 

36.  .r  +  y  +  z  =  9, 

a;  +  2y  +  4z=15, 
a;+3y  +  92-=23. 


41.  a;  +  2y-z  =  4.G, 
y  +  2z-a;=10.1, 
z  +  2a;  — y  =  5.7. 

42.  a;  +  2y-0.7z=-21, 
3a;  +  0.2y  — z  =  24, 
0.9a;  +  7y-2z--2'7 

43.  .T  +  y--=.l|z  +  8, 
y  +  z-2|y-14, 
2  +  a;  =  3|a.'  -  32. 

44.  ia;  +  iy+j2=--3Gi, 

i^-  +  iy  +  i^  =  27, 

i^-  +  i3/4-|2=18. 


20G 


SIMULTANEOUS    LINEAR   I':QUATI0NS. 


45. 


46. 


47. 


48. 


X  -f  J.  ^  r, 


z+1 
3y+z 

3z_-f-_a;__  o 

y  +  1  " 


2, 


y-z 

aT  +  5 
a;  +  3 

53.  (a:  + 

(2/  + 

54.  (2x- 

(y- 


=  10, 
=  9, 
=  1. 

=  2, 
=  1, 


.49.    '-l^l, 
X      y 

9        Q 

-2-1  =  0. 

2/        2 

50.    -  -j h  -  =-  4, 

X     y      z 

3,8,5      . 
-  +  -  +  -==4, 

9  ,12      10      , 

X      y        z 

51.  -a.  =  i, 

.r  +  3/      5 

.7-  +  2        6 
2>r      __  1 

2  -j-  o;      7 

52-   7-^  =  20, 
4y--3a; 


10 


2a;-3z 


^ 


15, 


12. 


2)(2y  +  l) 
2)(32  +  l) 

l)(2  +  2)  = 

-i)(y  +  i) 

4)(z  +  l)  = 
2)(2  +  S)- 


4?/  —  52 

=  (2a:+7)y, 
=  (a; +  3)  (32-1), 
(y  +  3)(2  +  l). 

=  2(r.  +  l)(y-l), 
(a; +  2)  (2 +  2), 
(y-l)(2  +  l). 


1. 

4. 
0. 


-4, 


5 


-4, 


-12=4. 


1 

— > 

5 

1 

— > 

6 

1 

7' 

=  20, 

- 15, 
-=12. 

1), 


■%'V'..  '     I, 


simultanp:ou.s  linear  p:quationg. 

207 

55.    (:.  +  l)(5y-3)_(7.t'-f-l)(2y    -8). 

(4..--l)(2+l)--0f+l)(22-l), 
(7/  +  3)(2  +  2)-(3y-G)(32-l). 

56.  21.^•  +  31y-f  42z=115, 
G(2.r+y)  =  3(3.T  +  2)  =  2(y  +  2). 

57.  15(.r  -  2y)  =  5(2.^-32)  =  3 (v/  +  c), 
21.r-f- 312/4-41^-135. 

58.  (jx(j/-\-z)  =  ^i/(z  +  x)  =  3z(x  +  2/), 


'-  +  W-     9. 

0-'       ?/       z 

. 

59.    3  a:  4-y  +  z^-- 20, 

60.    x  +  z-{-Si/  =  BO, 

3w-f-a;+4y-30, 

5  ?«  -|-  y  +  2  =  10, 

3u-\-Gx  +  z  -40, 

4?^-f-:t  H-z-=  10, 

5w  +  8?/  +  32-50. 

3?^  -{-  X  -\- 1/-—  10. 

§  45i  The  principle  of  symmetry  is  often  of  use  in  the 
sohition  of  symmetrical  equations.  For,  from  one  relation 
which  may  be  found  to  exist  between  two  or  more  of  the 
letters  involved,  other  relations  may  be  derived  by  symme- 
try ;  also,  when  the  value  of  one  of  the  unknown  quantities 
has  been  determined,  the  values  of  the  others  can  be  at 
once  written  down,  etc. 

Examples. 

(•^  +  y)(y  +  2)  =  ^. 

(x-\-  z)(i/-hz)  =  c. 

Multiply  the  equations  together  and  extract  the  square 

root. 
•••  (^  +  y)(y  +  z)  (z  +  x)  =  ^(ahr). 


208 


SIMULTANEOUS    LINEAR   EQUATIONS. 


Divide  this  equation  by  the  third. 

.\x-\-y  —  ^^ ^  ;  and,  therefore,  by  symmetry, 

c 

■\/{ahc) 


y-\-z 


a 


(1) 

(2) 
(3) 


0 

Hence,  we  get 

ab  —  bc-\-ca 

^         2^{ahc) 

whence  y  and  z  may  be  derived  by  symmetry. 

2.  a:  +  y-fz  =  0, 
ax  -f  hy  -\-cz  =  0, 
box  -[-  cay  +  abz  -\-(a  —  b)(b  —  c)  (c  —  a)  =  0. 

cX(l)-(2)   gives  (c-o)x-\-(c-b)y  =  0. 

Hence,  y  =  ^- '-^,  and  similarly, 

b  —  c 

{a-~b)x 

b-c 

Substitute  in  (3)  these  values  of  y  and  z,  and  reduce ; 

then,  x{a  —  b)  {c  —  a)  =  (a  —  b) (b  —  c){c~  a), 

ov,     x  =  b  —  c. 
Hence,  y  =  c  —  a,  z  =  a  —  b. 

3 .  a(3/z  —zx~ xy)  =  b{zx—xy— yz)  —  c{xy ~yz~ zx)  —  xyz. 
Divide  the  first  and  last  equations  by  axyz ;  then, 

;,  and  hence,  by  symmetry, 


a 
1 
b 

X      y 
_1      1 

?/       z 

1 

X 

1 
c 

1       1 

Z        X 

1 
y 

SlMULTANEOrS    LINEAR    E^rATIONS. 


L'09 


Therefore, 


0       c 


o 


— -,  und,  by  symmetry, 

X 


c      a  y 

1  ,  1_       2 

a      b  z 

4.  ax  -\-hy  Ar  cz  ~=  1,  (1 ) 
a'x-{-bhj-^ch^l,  (2) 
a'x  +  bhj  +  c'z^l.  (3) 

6-  X  ( 1)  —  (2)  gives  a {c  —  a) x  +  6 (c    'b)y  =  c      1 .    (4) 
6'X(2)-(3)givesa'(c-a)a;  +  ^'(c-^)y--6'-l.    (5) 
i  X  (4)  —  (5)  gives  ai(<?— a)a;— a'^(c— a)ar— J(6'— 1)— (c— 1), 
or  a(a  — 5)(a  — c):r  =  (c  — 1)(Z>  —  1). 

Iherefore,  x  =  -^^ :^- \  I 

a(a  —  b){a~  c) 

whence  y  and  z  may  be  derived  by  symmetry. 

5.  Eliminate  x,  y,  z,  u  (which  are  supposed  all  different) 

from  the  following  equations : 

X  —  by  -\-  cz  -\-  da, 
y  =  cz-\-  du-\-  ax, 
z  ^=  du -\- ax -\-  by, 
u~.ax  -{-by  -\-  cz. 

Subtracting  the  second  equation  from  the  first, 

x  —  y^=by~  ax,  or  (1  +  a)  a;  =  (1  -f-  b)  y, 

which,  by  symmetry, 

-={l-\-c)z  =  (l-{-d)u. 

These  relations  may  also  be  obtained  by  adding  ax  to 
both  members  of  the  first  equation  by,  to  both  mem- 
bers of  the  second  equation,  etc. 


210 


SIMl'LTANEorS    LINKAU    EtiUATlUA'.S. 


Isow  divide  the  first  equatit^u  Ijy  these  equals. 


+  a 


d 


1-l-i      1 


\-\-d 


And 


KUieo 


\-\-a 


^1- 


(t 


1-1 


we  liave 


a 


1  = 


a 


1  +  a      1  +  ^^      1-1-6-      \-\-d 


Ex.  57. 


1.    Given  ax  -|-  />v/   -  c^ 
a'.c  -f-  Z>'?/  •—  a', 
Given  bx  --  ay, 


and  that  x 


h'c  -  hc^ 


2. 


and  that  x 


a  -  ■  ha' 
derive  the  vahie  of  y. 

_  a,  {dm  -    rn) 
be  —  ad 
dx-{-indr^cy-\-nd,    derive  the  vahie  of  y. 

3.  Given  ax  -[-  hy  +  cz  —  d,     and  that  x  =  -^^ -~^ — — S 

a  (a—b)  (a—c) 

d^x-^-h'^y-^-c^z  —  d'\    write  down  the  values  of 
a^x-{-b^y'\~c^z  =  cZ'\    y  and  z. 

4.  There  is  a  set  of  equations  in  x,  y,  z,  u,  and  w,  with  cor- 

responding coefficients  (a  to  x,  etc.),  a,  h,  c,  d,  and  c ; 
one  of  the  equations  is  x  =  by -}-  cz -j~  du  -\-  eiv,  write 
down  the  others. 

Solve  the  following  equations : 
X    .  ?/  y  .  z       1      X 


' 


7;i      ?i  ?i     2') 


c. 


b,  -1.+- 

6.  a;  +  «y  +  ^2  =  7>i,  y  -j-  az  -}-  hx  =  7i,  z  +  (7.r  -f-  by  ^j)- 

7.  x-\-  ay  =■!,  y  -f-  i^  =  ?^i,  2;  +  ew  =  n,  ih-\-  dw  =  p, 

t(;  +  eo;  =  r. 


SIMULTANKOrS    LIN?:AR    IHiUATIONS. 


1211 


Is. 


c  —  he' 
J,  -  -  ha' 
3  of  y. 

'dm  -    cti) 
he  —  ad 
3  of  y. 

[d-h){d-e)^ 
a—b)  (a—c) 
values  of 


w,  with  cor- 
e,  d,  and  e ; 
-f-  ew,  write 


\w  =p, 


8.  Eliminate  x,  y,  z  (supposed  to  be  all  different),  from 

the  following  equations:   x~bi/-{-cz,   i/=^cz-{-ax, 

z  =  ax-\~  hy. 

9.  Eliminate  x,  y,  z  from 


— —  =  a,      -'-  -  =-  0, 

y-\-z  z-\-x 


-_z  c. 

x-\-y 


10.    Having  given 


X  ^^hy  Ar  cz  +  du  -f  c^o, 
y  ■—  cz  -\'  du -\-  eio -[-  ax, 
z  =  du  +  CIV  -f-  ao:  -f  hy^ 
u  =  c?^  -{-  ax  -{-  hy  -{-  cz, 
w  =  ax  -j~  hy  -{- cz  +  c?ii) 


d 


show  that  ^- (-  7-7^  +  T~; h :; + 


1+a      l+b      1+c     1+d     1+c 


-1. 


§  46.   Resolution  of  Particular  Systems  of  Linear 

Equations. 

Examples. 

1.  x-}-y-{-z^a,  (1) 

y  +  z  +  u^-h,  (2) 

z  +  u-{-  x  =  c,  (3) 

w+^-hy  =  c?.  (4) 

(l)+(2)+(3)+(4)   S(u+x+y+z)  =  a-{-h-{-c  +  d,    (5') 
3(1)  3(a;+y  +  2)  =  3a,  (6') 

i[(5')-(6')]  2^-i(-2a  +  ^  +  ^  +  cZ). 

The  values  of  x,  y,  and  z  may  now  be  written  down  by 

symmetry. 
The  following  is  a  variation  of  the  above  method,  appli- 
cable to  a  much  more  general  system. 
Assume  the  auxiliary  equation 

u-\-x-{-7/-^z  =  s.  (5) 


SIMULTANEOUS    MNKAR    KlilATlUNS. 


2. 


3. 


Hence,  (Ij  becomes  s    -  w-- a,  (G) 

(2)  becomes  s  ~  x  =  h,  (7) 

(3)  becomes  s  —  i/  =  c,  (8) 

(4)  becomes  s~  z  ~d.  (9) 

(5)  +  (6)  +  (7)  +  (8)  +  (9),  is=^^s-ha  +  h-i-ci-d. 

Therefore,  s~^(a-{-b-^c-{-  d). 

s  is  now  a  known  quantity,  and  may  be  treated  as  such, 


in  (6)  giving 

U  ~=  iJ     -  Ct, 

in  (7)  giving 

x  —  s~  h, 

in  (8)  giving 

y~-s~c, 

in  (9)  giving 

z  —s~d. 

yz-a{y  +  z), 

zx  —  b(z  -}-  x), 

xy-^c(x  +  y). 

(1)  :  a^jz, 

1      1      1 

y      2     a' 

(2)  -:  hzx, 

1    11^ 

z     X      b 

(3)  -^  cxy, 

1      1      1 
X     y      c 

(1) 

(2) 
(3) 


This  may  now  be  solved  like  Exam.  1,  using  the  recip- 
rocals of  a,  b,  c,  X,  y,  and  z,  instead  of  these  quantities 
themselves. 

aiu  +  bi(x -\- y  +  z) '- 
a^x  i- b^(y  +  z  +  u) -- 

(hy  +  ^3  (2  +  w  +  x)  '- 
a^z-\-b^{U'\-x-^y)-- 

Assume  the  auxiliary  equation, 
u-\-x-\-y-{'Z=^s. 
(1)  becomes  b^s  —  (bi  —  ai)u  =  c^. 

b\         ,  Cy 


Cu 

{^) 

<7y, 

(2) 

^3, 

(3) 

6'4. 

(4) 

(5) 


Therefore, 


bi  —  ai 


u  = 


*. 


a, 


(6) 


i 


4. 


SlMlI/rANK»)r.->    IJNKAll    l.tilATIoNS. 


(«) 

0) 

(S) 

m 

+  e 

+d. 

ited 

as  suuli, 

(1) 

(2) 
(3) 


Ing  the  recip- 
lese  (luantities 

(1) 
(2) 
(3) 
(4) 

(5) 
(6) 


Siiiiilarly,  iVoiii  (- ), 


Similarly,  Iruui  ('>), 


/; 


.s       .f 


>o  >t., 


h.,         (I., 


h. 


s~!/       ,     — 


^3         «3 


Similarly,  from  (1),  - — * —  .s  —  s  --  — 


1u~ 


(Ia 


K- 


(U 


(<) 


(S) 


0') 


(5)  lOi)  |-(T)  +  (8)-K^). 


h 


V 


h 


bx  —  «,       /-».,  — 


-h 


i- 


h 


('; 


•:,  -^^.,        />,-    (it 


s       s 


h 


+ 


h  —  ax 


CU  (K  On  h.  (I 


I- 


MO) 


From  (10)  wo  can  al  oikjo  get  the  value  of  .s',  which  may 
therefore  be  treated  as  a  known  quantity  in  (G),  giving 


u  ~ 


bi  —  ai  ■    . 

and  the  values  of  x,  y,  and  z  may  be  o])tainod  from 
(7),  (8),  and  (0),  or  they  may  be  written  down  by 
symmetry.  . . .       ■ 


4. 


ax  -h  b  (y  +  2)  c, 
ai/-\-b(z-\-u)  -^d, 
(iz  +  bl^ii-\-  x)  —  c, 
aiL+b{x-[y)  --  f. 

u -{- X -\- y -\- z  i=- .S-, 


(1) 
(?) 
(3j 

(4) 
Assume  \t -{-  x  -\- y  -\-  z  -  «,  (5) 

(l)-K2)  +  (3)  +  (4),  (a  +  2/>)s  -c-h^Z-h6-f/  (B) 

Hence,  s  is  a  known  quantity,  and  may  be  treated  as 
such. 

From  (1)  and  (5),  bs  —  bu  +  {<(■  —  b^x  ■=-  c. 


Therefore, 


bit  —  [a  —  b)x  :—  bs  —  c. 


Simihirly,  from  (2)  and  (5), 


bx  -—  (a  —  b)  y    -  bs  —  d. 


0) 


(8) 


214 


SIMULTANEOtlS    MNKAR    EQUATIONS. 


5. 


6. 


From  (3)  and  (5),  by  {a  —  h)z  =  1m  c. 
From  (4)  and  (5),  hz  -  (a  -  b)u^bs  -  -/. 
bO)  +  (a-b){S), 


'u 


(a  —  byi/  --  aba    -  be  —  (a      b)  d. 


6(9)-h(«-/;)(10), 


(10) 

(11) 

(12) 


i'y  —  (a  —  byu  =  abs  —  be  —  (a  -  b)j. 
b\\\)^{a~  b)\\2\ 

[b'~{a-by]u     abs[b'-\-{a~by]    a[b'd^a~bYf] 
~-b[b\c-^d)^\-{ct'-hy(^e-f)\  (13) 

The  values  of  x^  y,  and  z  may  now  be  written  down  by 
symmetry. 

a'  -}-  a?x  +  «?/  +  2  ^  0, 
b^  -\'b''x-\-hj-]-z  -  -  0, 


-f-  c^x  -[■  cy  '\-  z      0. 


a, 


The  polynome  ^  -\- xO  -[- yi -\- z  vanishes  for  t 

t  =  c. 
Therefore,  by  Th.  II.,  page  58,  for  all  values  of  t, 

f  ■\-  xe  +  yt^  z  -=  {t  -  a){t  -  b) (t  -  c) 

=  i^-(a-{-b-\-c)t:'-{-(ab'^bc-\-  ca)t-  abc. 

Therefore,  by  Th.  Ill,  pape  67, 
a;  =  —  (a-f-  i  +  c), 

y  =  «^  +  ic  +  ca^ 
z  =  —  abc. 

^  +  y  +  z  +  w  =  l, 
ax  -\- by -\-  cz  -\-  du  =  0, 
a'x  +  b^y  4-  ch  +  d^u  =  0, 
aJ^x  +  Fy  +  c^z  +  d'2t  =  0. 

Employing  the  method  of  arbitrary  multipliers, 

(4)  +  ^(3)  +  m(2)  +  «(l). 


t  =  h. 


(1) 
(2) 
(3) 
(4) 


a' 


+  Id' 
-{■ma 
-\-n 


x-\-  P 

y-\-  c^ 

z  +  d' 

+  lb' 

-\-lc' 

+  ld' 

-{-mb 

-}-mc 

-\-vid 

+  71 

-\-n 

-\-n 

u 


=  71         (5) 


(9) 
(10) 

d. 

(11) 

)/. 

(12) 

MV{a 

(13) 

ben  do^ 

^vn  \)y 

les  of  t, 


—  ahc. 


(1) 

(2) 
(3) 
(4) 


Uers, 
u  —  n      (5) 


SIMlLTANEOUrt    LINEAR    EiAl'ATIONri.  ^IT) 


To  determine  a\  aHHumo 

&  -f-  Ic^  -\-mc  4-n    -0, 


Thoretbro, 


.r  ~ 


a'  -f  (?«'  -|-  nvi  '\-  n 


(8) 
(0) 


But  the  system  (0),  (7),  (8)  has  been  solved  in  PiXiim.  T), 
IVoin  which  it  is  seen  that 

l  =  —  (b  -{-  c-\-  d),  VI  ~  bo  -\-  cd  -\-  dh,  n      —  bcd^ 

and  <i^  -\-  iH  -\-  am  -\-  n  -=  {a  —  b)  (a  —  c)  (a  —  d). 

Hence,  using  these  values  in  (9), 

—  bed 

X  = 

(a~b){a  —  c){<t  ~d) 

The  values  of  y,  z,  and  u  may  now  be  written  down  by 
symmetry. 


7. 


X 


+ 


y 


+ 


in  —  a     7)1  ~  b      rti  —  c 

X 


+ 


y    ^    z 


n  —  a     71  —  b      n  —  c 


ij) 


X 


Assume 


P 
1 


a 


+ 


y 


X 


p  —  O       J) 


y. 


-1, 

(1) 

-1, 

(2) 

=  1. 

(3) 

t  —  a      t  —  b      t~c 

f  +  Bf  i- a -}- 1) 


(4) 

But  in  virtue  of  equations  (1),  (2),  and  (3),  the  fu\st 
member  of  (4)  vanishes  for  ^  =  m,  ^  =  7?,  and  t=2^'y 
and  hence,  f  +  -^^^  -{-Cl-{-  I)  vanishes  for  the  same 
values  of  i;  and  therefore,  by  Th.  II.,  page  58, 

f  +  Bt''-\-Cti-D=^(t~m)(t-  7i)  (t-p). 


1^10 


SIMT'T,TAN7':nT-3    LINEAR    EQI'ATIONS. 


Therefore,  (4)  becomes 


1    - 


_  _z_ _  {t-~m){t'-n){t~-p) 
t-:t      i-b      i  —  c'  {t-a){t~b){t~    c) 


V 


To  obtain  tbe  value  of  .r,  multiply  both  sides  of  this 
■       equation  by  (^  —  (i). 

•       '       '      '  t—h  t-c  {t-b){t-c) 

Now  i  may  have  any  value  in  this  equation ; 
let  t=za.  ' 

Hence,  x  =  (a-m)(a-n)(a-p)_ 

[a  ~b)ia  —  c) 

The  values  of  y  and  z  may  now  be  written  down  by 
symmetry. 

8.    x+_a^yVh^^z±j^^      ...  . 

p  q  r  • 

Ix  +  7?iy  +  7iz  -=  s^  (2) 

By  §  37, 

X  -\-  a  _  ?/  -f-  h  _  z-\-  c  __  Ix -|- tny -\-nz-\~  la -\-  tub -\- 71c 
2?  q  r  Ip  -\-  7)iq  +  7(.r 

/ON  «^  +  Iti  4-  7nb  4-  no       -n 

(2)  =      !     ,     — -^— =  i?,  sa-; 

therefore,       x^pR  —  a,  y ~qR~-b,  z  —  rR-c, 


9.   yz  -\-  zx  -f  xy^=^{a-\-b-\-  a)  xyz. 
7/z  +  ZX  __  ZX  -}-  xy  __  xy  -f  yz 


a 


c 


(1)  -^  xyz,      -  H 1-  -  r^  a  +  ^  H-  ^.  ' 

^  ^         •         xyz 


(2)  ■--  .iv/s. 


i+i  i.f-1-  V^ 

.'f      y     7/      z      z      X 


a 


b 


(1) 

(3) 
(•1) 


){t-    c) 
ides  of  this 


i; 


en  down  >)y 
(2) 


nr 


—  rR  —  c. 


(1) 

(2) 
(3) 

(4) 


10. 


SIMULTANEOUS    LINEAR   EQUATIONS. 


217 


O 


§  37  and  (3), 
(4)  and  (5), 


A- 


—  —  ^j 

y 
1 


a. 


-  +  -  + 
^x      y 


-\~~  -^2h 


—  Q 


+ 


o 


SC. 


X 


(3)~(G),  -^a~h-\~n, 


X 


y 


a 


\h 


-ffl 


X 


[-0  ,  y^-h 


. ._  9 


X  ~h      y  —  c 


a  —  c 


(1). 


-  + 


a 


a  -\-h 
X  —a 


1=1-^-+^. 


2/ 


«-]-  c 


a-\-h 
Similarly,  from  (2), 

x  —  a  —  h-\-c  __a  —  h-\-c—y 


a 


a 


(3)  and  (4), 


a 


+  b 


X   -  a  —  h-\-c  =  — - — :(«  -rh  -[■  c  —  y) 


a  —  c 


a 


a-\-  c 
(a    -  b-i-^c    -y). 


But,  unless 


a 


a-{-  c 


a 


a 


-b 


this  cannot  be  the  case  except  for 


a 


h^- 


y 


=  0. 


in  w 


Inch 


case  X  —  a 


b-\- 


Oal 


so. 


giving     X 


a 


0  —  c  anc 


a 


h-\- 


If 


a 


+  ^_ 


a 


a  -f-  c 


a 


-b 


a 


I 


'r  =  (I  —  c 


P  ~  c^  ^  0,  or  (b  -[-  c)  {b  -  c)  -.  0. 
Therefore,  b  ~  c  ov  b  —  —  c. 


(0) 

(1) 
(2) 


(3) 


(4) 


(5) 


218 


SIMULTANEOUS    LINEAR   EQTTATIONS. 


But,  if  h -- -\- c  or  —  c,  (1)  and  (2)  are  one  and  the 
same  equation  ;  hence,  if  (1)  and  (2)  are  indepen- 
dent, (6)   cannot  be    true,   thus   leaving   only   the 


alternative  (5). 

11.  2ax  =  (h  -\-c  —  a)  (?/  +  z), 
2  bi/  ==  (c  -{-  a  —  h)  (z  -[-  x), 
(.T  +  2/  +  zf  -\-x''-i-if-\-z''=.A(a''  +  V'  +  c'). 

(1)  and  page  155,  (5), 

X      ^y  +  ^^^^  +  y  +  g 

(2)  and  pago  155,  (5), 

y       _  ^  +  2  ^  ^  +  y  +  ^ 

c-\-  a  —  h        2b        c-\-  (i-{-b 

(4),  (5),  and  page  155,  (5), 

.  X  +  ?/  +  2? ._        X ^ z 

a-\~b  -\-  c      b-{-  c—-  a      c-\-a  —  b      a-\-b  —  c 


(1) 
(2) 

(3) 


(4) 


(5) 


{b^c-  of 


(■T  +  .y  +  ^y  +  :y^  +  ?/  +  2^ 


{a-\-b  +  cy-\-{b-\rC~-ay-\-{c+a-by-^(a-\-b-~cy 


Reduction  and  (3)    ^  0^'  +  y  +  zf  +  x"  +  f  +_ 
Therefore,  x^  —  {h  -\-  c  ~  cif. 


-1. 


1 .  ax  -|-  by  —  c, 
mx  -\-  ny  —  d. 

2.  ax-{-by  =  c, 
mx  —  mj  =  d. 


Ex.  se. 


3.    ax-\~by=^c, 
inx  -f  ny  —  c. 


4.    f  +  ^ 
a      0 


1, 


■r  4-  y  -^^  c, 


SIMULTANEOUS    LINEAR    EQUATIONS. 


219 


ine  and  the 
re  indepen- 
g   only   tlie 

(1) 

(2) 
).  (3) 


(4) 


(5) 


b-c 


(a  +  6-ey 


5. 

a      h 

f+^   1. 

0      a 

• 

6. 

x^U     1 

a      b 

X     y^ 
b      a 

7. 

ax  -\-bc  —  by-\-  ac, 

x-{-y  —  c. 

8. 

a      b 

=  m, 

X     y 

b      a 

n. 

X     y 

9 .    {a-\-c)x  —  (a  —  c)y  =  2ab, 
(a-\-b)y — (a  —  b)x  —  2ac. 


10. 


X  —  c     a 


y 


-c      b 


X  —  y  —  a 


-b. 


11. 

X      a 
y     ^' 

X  -{-m      c 

yi-71      d 

12. 

x-\-y      a-\-b-{-c 
y-\-\      a  —  b-\-c 

y—l     a—b—c 
x-{-l      a-{-b-~c 

13. 

x  —  a-{-  c      b 
y  —  a-j-b      c 

y-\-b      c-\-a 

x-\-c      b-\-a 

14. 

x^c      y-Vb  _cy 
a-\-b      a-\-c 

X      b      y      c      c_ 

a 


a 


-b 


15. 


X 


tn  —  a 


+--^,=1 


TTl 


~b 


X 


n  —  a 


+ 


1. 


n 


-b 


1. 


16.   x-{-y-\-z  =  0, 

(6  +  c)n:  +  (a  +  c)y  +  {a  -\-b)z  =  0, 
box  +  O'Cy  +  (ibz  —  1. 


—  c. 


17.    X  -{-y-\-z  =  I, 
ax  -\-by-\-cz  — 


111, 


X 


a 


+ 


y 


+ 


i-b  '  I- 


220 


filMULTANEOl'S    LTNKAR    EQUATIONS. 


It 


18. 


X      a      y  —  h      z  ~  c 

l{x  —  a)  -\-  m  (?/   -  h)  -f-  n  {z  —  c)  =  1. 


.19. 


X  ■—  a      y  —  h 


')      z  —  c 


'p  q  r 

Ix -^^  my  -^i- nz  --=  1. 


20.    (t{x  —  a)  =  h (y  —  h)  ■.-  c {z  -  c), 
ax  -\~hy-{-cz~  r/i\ 

21!.    X '[-  y  -|-  z  — -  rt  +  i  -f  c, 

l)X  -|-  cy  -f-  az  =-  (t^  +  h'^  -\-  c\ 

r.v   \-  ay  -[-  hx  ^-^  (i^  -\-  Ir  -}-  r^. 

22.  ,;/•  \   ,i-\-.z^a^h^c, 

a.r  -  \-  by  -f  cz  ---=  ah  -\-bo-\~  ca, 

v/>  —  c)x  h  (c  —  a)y  -]-(«  —  h)  z  ~0. 


23.    j:  \-y  \-  z      m, 
X  :y  :  z  ---  a\h  :  r. 


24.    ax  -\-  hy  -{-  cz  =  r, 
mx  =^  ???/,  qy  --jiz. 


25 .    xy  '\-yz-\-  zx  —  0,  ayz  -f  hzx  -\-  cxy  =  0, 

hcyz  -h  acxz  - 1-  aZ>.i^?/  +  («  —  Z>)  (5  —  c)  (^  -  rr.)  xyz  —  0. 


26.  (ri  +  ^>)-'?'  +  (^  +  c)y  +  (^  +  «):z 
{a-\~c)x-\-{a-\-h)y-\-(1>-\-c^z 
{h  \  c)x  -\-  {a~[-  c)y -^  {a-^h)z 


~  ah  -\'bc-\-  ca, 
—  ab  -f  «(?  +  be, 
=  a'  +  //  !-c^ 


27.    'tnx -\-7i'y -[- pz-]- qu 
X   _y 2  _  w 


1. 


1 


ftiMrr/rAXEoi's  linear  EtirATioxs. 


001 


28. 


^  (y  +  -)  _  ?/  i^'  +  -)    2  (x  +  ?/) 


(/ 


29. 


(a  —  i)  (.r  -[-  c)  —  a?/  +  hz  —  (r^  —  a) {y-\-h)  —  cz-\-  ax 


-0, 


i,> 


1=0. 


cz  =  ■>■ 


b?/  =792. 


n!. 


)  .^2/2  ^  0. 


4-m, 


<? 


30.  «.?;-I-%— 1, 
i?/  -f-  ^-  ^  1, 

cz  +  rt.r  =  1. 

31.  ly  +  mx  -—  ??, 

'n\%\ny  =  ^. 

32.  .r  +  ?/  — a, 

?/  +  2  =  />, 
r?.-  +  2  =  (?. 


33.    y  -f  2  —  X'- 


nnn 


-{-x-y  -- 


a;  +  2/  — 2 


In 


7)1 


hn 


n 


34. 


1    .    1 


f  -  =  2 


f'. 


y 

i  +  ^-^2.. 


1      .     1 


+ 


y 


\LC 


3B..  1  +  1-1 

2/      z      x 

i  +  i-l 

z       X      y 

X      y      2 


— » 
(t 

o 

2 

c 


36.    (rt4-i)a:+(rt-Z»)2=-2^'r7, 
(/> +6')  3/ 4-(i— c)  a:  =  2a(^, 


37.    x-\~'j-~ 


a. 


y-\-- 


=  k 


a 


f 


^  _y 

a      h 


~-  c. 


38. 


X 

L 

.7 

X 

2 

y 

-i- 

-  =  ?>  -  - 


a, 


=  a  —  c. 


^.'  +  a      a  I  /> 


222 


SIMULTANEOUS    LIxVEAR   EQUATIONS. 


39.    x^y 

—  z  "'  a, 

40.    n  -\-  V  —  X  —  a, 

y^-z  - 

—  V  -^  b, 

V  -{-  x  —  y  =  b, 

Z  '\~  V  - 

-  X  —  c, 

X  -\-y  ~z  —  c, 

V  -\-  X  - 

-  y  -  ^• 

Ex. 

y  -\-  x  ~  u  ~  d. 
z  -{-u  —  v  —  e. 

59. 

Resolve : 

1.    {a-\-b)x 

+  {a 

-b)y^2(a^  +  b^), 

(a^ 

b)x 

i-(a 

i- b)y- 2(0" -b'). 

2.  xi-y=a, 
x^  —  y^  —  b. 

3.  2x  —  2>y  ^=in, 

2x^  —  3?/  —  n^  -f  xy. 


4.    (a—b)x-{-(a-\-b)y=a-\-b, 
a-\~b      a  —  b      a-{-b 


5.    (a  ~-  b)x4-y=^ — ' — . 


x-\-{a-{'b)y 


_a  —  b-\-l 


a 


6.    {a-{-b  —  c)x  —  (a  —  b-\~c)y  —  4:a(b  —  o), 
X  _  a  -\-b  —  c 


7. 


8. 


y    «^ 

b-^c 

x-]-y  _     a 

9. 

X     y  +  ] 

L 
—  /» 

x  —  y     b  —  c 

X  -  y  —  ] 

L      ^■ 

oo-\-c     y  -\-b 

x-^y-\  ] 

L      i 

a-{-b      a-\-  c 

a:  +  y  -  ] 

—  u, 

[ 

X  —  a      a  —  b 

10. 

•^  +  .y  +  l 

.      a  +  1 

y  —  a     a-^b 

X      y  +  1 

.      a-1 

X      a'      P 

ar  +  y-f  1 

■      l  +  b 

y      d^  -{-b^ 

ar  -  ?/  -  ] 

.      1-6 

SIMTLTANEOUS    LINEAR    EQUATIONS. 


18.  {((  h){x-\-c)-oi/[-hz^O, 
{c-~a){y-\-h)—cz^ax~Oy 
x  +  y-\~z  =  2{a-^h-{-c). 

19.  :^^^^-{--l—=a-\-h, 


h  -\-  c      c  —  a 


- ,      x  -  y  +  1 

11.  ^.^L^^a, 

x-y~-  1 

12.  -^  +  -^  =  «  +  ^^, 

^•  +  y-=2«. 
a      0 

13.  {a-{-c)x-\-{a—c)y  =  2ah, 
{a'\-h)y—(ci~h)x~2ac.    20.       ^ 


IL 


+ 


c  -f  «      a  —  i 


^  +  ^, 


+  ■; ~  c-\-a. 


14.  a'^  +  aa:  +  2/  =  0, 
i'-'  +  ia;  +  3/  -=  0. 

15.  y-\-z~-x~a, 
z-\-x~y  =  h, 
X  -\-y—  z  =  ^. 

16.  1  x-{-l\y  -{-  z  =  a, 
ly -\-l\z  -\- X  =  h_ 
7z  +  •laT  +  y=--c. 


1 ^_^o. 

=  0, 


b-^-c     c  —  a     a  —  h 

^  L4.„_l_ 

h~c     c~a     a-\-h 


X 


JL 


i  + 


c     c~a     a 


+  -''--2a. 


+  h 


21. 


a; 


+ 


J/ 


.-  + 


a      a  —  1      a  —  2 


X 


-\ 


IL 


+ 


&      i-1      ^»-2 


X 


-4- 


X 


l^c-2 


17.    ^  +  ^-£  =  2a5 


a; 


y      z 
c      a 


-  + --2ie, 

y     z      X 


c  .  a 


-+----2 
z      X     y 


ea.. 


22. 


y  +  2; 
z-\-  X 


a. 


-i. 


23. 


E  4-^4-?=^' 4-^-4-^=^-^2/ 


a 


+  ^  + 


1,1.1 


h      c      a       cab 


-f^  +  ?--  +  7+- 


a 


001 


SIMULTANEOUS    LINEAR   EQUATIONS. 


.  Hi. 


|,«j 

i 


24.    'I.^^? 

a         O        C 


u 

1 


m      n     p      q      r 


25.    a37  =zhy^=^cz  —  (hi, 


if  —  ^  —  x~  u 

26. 

y  --Z  -^  an, 

X  -j-  z  —  i/i,      . 

a;  -j-  y  —  cu, 

1  —  X     a 

27. 

X  -j-  y  —  m, 

yJ^Z  -^V, 

z  -{-21  —  a, 

21  —  X  —  ^. 

28.  llx-\~9y  +  z—u^a, 

lly  +  92  -f-  u  —  X  —  h, 
11  z  i-^u-\-x  —  y  —  c, 
llu-\-9x  -{-y  -    z  -^  d. 

29 .  x-\-ay-\-c^z-\-c^u-^a^  =  0, 

xMhj-^dh-\J?'iL\d''=  0. 

30.  X  +  y  =  a, 

y^z=-h, 
z  '\-  11-—  c^ 
u-\~  V  —  d,  . 
y  4-  a;  —  e. 


31. 

X  +  /y  ^  rt, 
z  +  ?i?^  —  e, 

u-{-pv  --  d, 

m 

V  -|-  (jx  —  e. 

32. 

r 

x-i-y  —  z  ^n, 
y-\-z-\-  u  -^  b, 
z  -\-  u  -^  V  ==  c, 
u-{-v-{-x-  d, 
V  -{-  X  -\-  y  —  e. 

33. 

X  —  y  +  z  — -  a, 
y  —  z  -\-u  —  b, 
z  —n  -\-  V  ■—  c, 
u  —  V  -\-  X  —  d, 
V  —  X  ■\-  y  —  e. 

34. 

x-\-y  -\-z  —u- 

-a, 

y  -\-  z  -{-11—  V- 

i. 

z-\~u-\-v  —  x  = 

~-c, 

11  -{-v-i-x  —  y- 

-<l 

V  -{~x-\-  y  —  z  = 

-  e. 

35. 

x-^-y-^-z  —  ii- 

-'v~ 

-a, 

y  -\-z  -\-u  —  v  — 

-X- 

-b, 

z  -\-u^v  —  X  — 

-y  = 

--C, 

u-\-v-^  x  —  y  - 

cZ, 

V  +  0-'  +  y  ~  z  ~ 

-u  = 

=  e. 

*fS. 


a. 


a, 

d, 
e. 


_^.SnrULTANEous    LINKAK    KunATION.S. 

36.      2.7;  — ■  ?/ -y   _J_    0,w  r. 

%  -2;  -  w  +  2?;-a;-^  3/;, 


9/v 

•^2   —  It 


2u 


2v 


37.    .-2..  +  3z^-2y  +  .^«, 

y  -  22  +  3:r  -  2w  +  ^;  ^  ^^ 
z-2^^  +  3y-2t;  +  :r  =  ^, 
w-2t;4-32-2a,--f  2/  =  ^.' 


a, 


2^ 

X 

y 

z 


=  a, 


I    s 
f     ' 


V 
X 


V 
X 

y 


=  a, 


y  ~  s  --^ 


d. 


u 


e. 


CHAPTER  VII. 


Pure  Quadratics. 

§  47.    (A)    If  an  equation  reduces  to  the  form 
(rnx  4-  ny  --  c'\ 

then  (mx  +  ^0^  —  c*  —  0. 

Hence,  (m^i  -}-n)  —  c~0,  and  therefore  Xi  = 

or  (mx2  -\-7i)  -j-  c==0,  and  therefore  Xj  = 


c  —  n 


c  —  n 


m 


(£)    If  an  equation  reduces  to  the  form 

^mar  +  riV a^ 

^px-i-qj~b''' 

,1                         qa  —  nb             —  qa  —  7ih 
then  Xi  —  -^— ,    X2  =  — ^ 

mo— pa  vib-j-pa 


c 


(See  Exams.  4  and  5  below.) 


Examples. 

J     .^•  +  3(a  — 5)_a(3.r4-0«-75) 
a;--3(a-^>)      5(3a7-7a  +  95)* 

Apply,    if  ~=-^,  therefore  ^"^^==-^  '"^- 
n      2'  m  —  np  —  q 


Hence, 


a; 


_  3.^'(a  +  ^)  +  9^'  -  14a6  +  9^' 

3  (a  -  ^>)  3  a;  (a  -  5)  +  9  (a^  -  6^) 


Dividing  the  denominators  by  3  (a  —  J), 

a:[a;  +  3(a  +  5)]  =  3:r(a  +  5)  +  9a'  -  l^ab  +  9^>^ 
Therefore,  a;"^  =  9  a'  -  14  a5  +  9  b\ 


c  —  n 


c  —  n 


m 


Lnd  5  below.) 


|a6  +  9i^ 


rUKK   (QUADRATICS. 


227 


2. 


Vr  f  4«  — 2V      5.f  h3a      9/>* 


Apply,  if  --=-?-,  therefore 

n       q  n 

and  factor  the  numerator 


?i  —  VI Q  —p 


P 


{x  -\-  4a  -  2hy  -  {x  -  2«  +  4i)- ; 

"       (x-]-4:a-2by       '    bx-{-Sa-\)/i 
.      x  +  a  +  b    _  x  +  4:a  —  2b  _     S(ab) 


■.r  +  4a  — 2^>      bx-\-Sa~db      ix-a-1b 

by  taking  difference  of  numerators  and  difference  of 
denominators. 

To  the  first  and  third  of  these  fractions  apply,  if 

m 


—  =^,  therefore 
n      q  n 


—  =-!—. 

7)1      q—p 


X 


+  a  +  &_       ?>{a-b) 


3(a-^)       4:?;  — 4a  — 46 
^[a^-{a  +  by]=^{a-b)\ 
x'=^\[^{a^by-\-^(a-bY]. 


^     {x+2Y__a 
x^  —  2x      b 


(:r  +  2y 


a 


(1) 

m  (x  -\-  2y  -\-n(x^  —  2x)      7na  -\-7ib 

But  (B)  can  be  applied  if  7.    md  n  are  so  determined 
that  m(x-}-2y  -}-n (x^  —  2 ^)  is  a  square. 

This  requires  that  4  m  (m  +  /i)  ==  (2  m  —  ny. 
.*.  4 w'^  +  4mn  =  4m'' —  4m7i -f  ^t^ 


228 


T'lIiK    qlAOUATICS. 


I 


ii 


f 


4. 


Assiiuic  vt,       I,  then  fi.-S,  and  (^Ij  bccoinuH,  on  Rub.sti-     ' 
till  ion  and  reduction,  | 


'  r\  say. 


(3.1;-  l3)'^""a-|-8^> 
•    '       3r      1        ''-  l-|-3r 


i^+Dl 


<(, 


a 


.         (a;'  +  2.r-Hy^ 

■ '  {x'-\-\){:x'-'ix  \-\y  h   . 

For  a;''  +  1  write  xz. 

•     {xz-^k-lxj  _  a        ,   (z  +  2)'  .__  a^ 
xz{xz  —  2x)      b  z{z—2)      h 

This  equation  was  solved  in  Exam.  3,  hence  z 
treated  as  known. 


may 


be 


But 


x^  +  1 


+  2:^-  +  l_.2  +  2 


^'- 


X 

1-1 


X' 


-2x^\      0-2 


2-1-2 

3-2' 


a  form  solved  in  {^B). 


Ex.  60. 

1 .  (.r  +  «  +  5) (a;  -  a  +  />)  +  {x -\-ii-h) {x -a-b)=--0. 

2 .  (a  -f  Zi.-f)  (/;  —  arr)  +  (^  +  c:^)  (c — i.r)  +  (c  -f  ar)  (a  —  ex)  ==  0. 

3.  (a  -\-  hx)  (ax  —  h)-\-(h-\-  ex)  (bx  —  c)-{-(e-{-  ax)  (ex  —  a) 

^^(^a'  +  b'  +  e'). 

4.  (.?.  +  x)(b-  x)  +  (1  +  «^^0  (1  -  ^^)  =  («  +  ^)  (1  +  ''^')- 

5.  (a  +  a:)  (5  +  a;)  (c  —  :c)  +  (a  4- a:)  (6  -  a;)  (c  +  a;) 

+  (a  -  a;)  (^  +  >'^)  (c  -{-x)  +  (a--  a;)  (5  -  x)  (c  +  .f) 
+  (a  —  ^)  (6  +  x)  {c  —  x)-\-{a^  x)  (b  —  x)  (c  —  x) 
=  5abc. 


\ 


IMRK    grAIiRATICS. 


OOf) 


on  siil)«ti- 


>ce  2 


may  be 


-  b)  =--  0. 

ax)  {ex  —  a) 

\h)(l  +  ^')- 

-  x) 

x)  (c  +  ^0 
\x)(c-x) 


6.  (a  -f-  a;)  (6  f-  .r)  (c  -f-  .r)  f  (it  +  j:)  (^  -f-  x)  (c  ~  x) 

I  4-  (a  +  x)  (b  -  .r)  (o-\-x)  -f-  (a  ~  x)  {b  f  ./•)  (c  -f  x) 

I  -{-la  +  a-)  (b  —  x)  (c~x)-{-  (a  -  x)  {b  \-  x)  (c     x) 

\  -{•(a  —  x)(b  —  x)  {c-{-  x)-\-  (rt  —  x)  (J)  —  x)(c~~  x) 

^%x\ 

7.  {a  +  56  +  x)  (5a  -f-  b -\- x)  ^  3(rt  -f-  /;  -|-  x)\ 

8.  (rt  f  17/>  -f  .r)(17a  -h  A  -f  .r)  -  9(a '\^b'\- x)\ 

9.  (9a  -  76  -f  3^) (0^*  -  7 a  +  3^-)  -  (3a  -f  3/>  +  x)\ 


11. 


12. 


13. 


14. 


ah 


cd 


10.  —li^-f— Ji^^ 

a:  —  a  ,  .!•  +  a      o 


=-0.    15. 


.Y  +  1      X  ~1 

a-\-x  _x  -\-b 
a~  X      X  —  b 

ax  -{-  b  _cx  ■]-  (I 
a  -f-  bx      c  +  dx 


a—  X 


1   -  b 


ux 


1- 


16. 


17. 


18. 


19. 


a  -  -  X  _  b  —  X 
1  —  aa:      1  —  bx 

xi-a-{-2b   J-2a'j~2x 
a;  ha-26      b-\-2a-2x 

a-\-4ib-\-x     Sb  —  a'\-x 
a  —  4ib-{-x      Sb-\-a~x 

x-{-6a-{-b  __  x~a~\'b 
x  —  Sa-{-b      a~-x-[-r^b 


a 


--1b-\-x_a-\-bb-\- 


X 


ax 


X 


la  —  b~x     5a4-6  +  a; 


20. 


21. 


22. 


23. 


24. 


3 a  "-  b  —  X  _  56  —  3 a  -f 


X 


a 


-36  +  x     5a-36  + 


X 


3a  —  26  -f  Sx 


X 


a 


+  26 


a 


26  + 


X 


3a; -3a +  26 


3a-26  +  3a;_.  ar— 7a  +  86 


a 


HI 


-26  + 


X 


3a;-  5a  +  46 


66  +  a;_3a-56  +  3 


X 


a-{-  X 


a 


+  6  + 


X 


a 


-\-b-x    ^  3  (g  -  6  +  x) 


3a  — 6  — 3a;       a  — 56+a; 


liSO 


rURE    QUADRATICS. 


33. 


34. 


25. 


26. 


27. 


28. 


29. 


30. 


31, 


32. 


/  ( 


t-\'h 


X 


_?y{<i,    -b   \-.c) 

5  a  4-  3Z>  —  3.6-      (t—  11  b-\-  X 

ba      h  -{-  X    _1[2a  —  h-^x) 
20r  '■  2/)-x)~~  a\-nb-x  ' 


lit—-b-\-  X a(a  +  5Z»  +  x) 

7 b  - a^x~ b{ba'\-b-\- x) 

X  -\-  a  ~h      a{x-\'-  a  -\-  bb) 
X  -  a  -\--  b      b(x  +  ba-\'b) 

'5 a  ~-ob-\-  x\     7 ti  —  0 b  -|-  3 X 


bb 


3  a-!-;?: 


7b      \)(t  I  3.^" 


a  — 


\~bb-\-x 


a 


+  l7b-{- 


X 


ba  +  b-\-xJ      17a  +  Z»  + 


X 


n 


a 


b'\-.vV_l1a-\-h-x 


\7b-a  +  xJ      11b-\- 


a  —  x 


1 


/  a  + 


b~ 


X     a 


{a-\-VJb-\-x) 


a 


+  llb~-x      b^iXJa^b-^-x) 


(x-\-la-^b){x-~a'\-b) 


X 


ba-\rb 


(:' 


)X 


!« 


^Ab){x-a+l1b)      bx-^1a-bS)b 


(^  +  3■l:  +  5■r^)(:r^  +  3.^•  +  5)  ^  9 
(1  +  2:f  +  3:f"0  [x'  +  2a;  +  3)      4' 


1 


\ 


35. 


-\ 


54  .r^ 


c'  -  lU-  4-  28      x'  -  17a;  +  70      .r^  -  14a;  +  40 


36. 


+ 


8 


a; 


v'~Q>x  +  b      o;^- 14a; +  45      0;=^- 11a; +  10 


37.    x\b  -  a')  +  a'(:r  -  b'')  -{-b^{a-  o;^)  +  abx(abx  -  1) 
=  (a-x')(b'-a*). 


QUADRATICS. 


L^;v 


). 


^X 


ijJ 


T 


\ 


{ 

i 

\ 


/ 


\ 


I 


/ 

iiuADRATio   Equations   and   Equations  that  can   be 

RESOLVED    AS    QUADRATICS. 


§  48.    ( ^■)    If  an  equation  appears  under  tlic  form 

(a-    .c){x      h)^-^c,  (1) 

tlien  .i\-^-Ua^\-h-\-r),  .1-,      l(a^f>    -i-), 

in  which  r'^  -'-  (a  —  h)-  -    4t'. 
From  the  identity 

(a  —  x)-{-  (x  —  h)=^  a  —  b 
we  get      (a--.^•)"  +  2(a-.^■)(.^'-7>)^-(.^•-  />y--(a--Z')".    (2) 
(2)  -4(1)  (a-xy  -  2  (a  -  x)  (./;  -  /;)  +  (x  --  by 
r  -  (jt  -  -  by  -  4c  — -  ?'^  say. 
Then,    [{a  -^  x)  -  {x  -  b)J  -  r^  --  0  ; 
lience,       [(a-:r,)-(.ri— i)]-fr  =  0,  and  .•..'•!-    ^(a-|-J+r); 
or,  [{a—x.^—{x2—b)'\—r  --  0,  and  ,".  x^  --    \{<.i\b    ?•). 


.^•'■ 


[^•  +  40 

To" 


1) 


2. 


Examples. 

./;'  +  {ab  +  ly^  --  {ii'  +  /yO(-^''  -h  1)  +  2(a'^  -     lr)x  +  1. 
. • .  x'  +  a'^i*^  -:  (tr  +  Z/'O  .y'  +  2  {a'  -  b'')  x  +  {a  -  /;)l 
.•.x'-\-  2abx' + a'^>''  --  (a + Z»)  V  +  2  (a'  -  b'')  x  +  (a-  by 
.'.  x^  +  a5  =  rfc  [{a  -]-b)x  -\-  {a  —  Z*)], 

or  x^  H=  (a  +  Z>)  ;r  +  «^  =  ±  (a  —  Z*). 
.-.  :f'^  q=  («  +  b)x  +  J(a  +  ^)'  --  i(«  -  by  ±  (a  -  -  />). 
,.  X  =f  i(a  -1-  b)  .^  I  -^[(a  -  by  i  4(a  -■  ^)]. " 

ax  +  ^  _  '>?^  —  '^i 
bx  +  a      ?2^'  —  ?/i 

Add  and  subtract  numerators  and  denominators, 

(a-\-h)(x-rl)  ^  (hi'\-7i)(x-l) 
(a-b)(x~l)      (m-n)(.6-+l)     • 


232 


QUADRATICS. 


\Xi 


'.!l±iy     (a-b)(m  +  n)  _  ^ 
^x  —  1/      (a  +  ^)  (^^  —  '^0 

5+1  S—  1 

s-1' 


=  5  ,  say. 


X2  — 


3.    (a  —  :ry  +  (^>  -  .1')*  =  c. 
In  the  identity 

(w  +  i;)*  =  2^*  +  ?;*  +  4  (iA  -f  r)^'i/v  —  2  wV 

let  w  =  a  —  a;,  v^=  x  —  h. 
.'.  ?^  + 1;  —  a  —  6  and  w*  +  v*  =  <?. 
.■.{a-by=^c^-^{a-b)\a~x){x~b)~2{a-x)\x- 

Write  z  for  (a  — a;)  (a:  — 5). 

.-.  z'  -  2 (a  -  ^^y^  +  (a  -  by  =  ^[c  +  (a  -  &)*]  =  ^^^ 
:.[z-{a-byY  =  i\ 

.•.hy(B),  z,  =  {a~by-t]  z,  =^  (a  -  by  +  t 
.'.  z  is  known. 

But  (a  —  x)(x~b)  =  z. 

•■■  by  (C),  xi  =  ■!-(«  -f-  Z>  +  r) ;  x,  =  |(a  +  ^>  -  r), 

in  which  r'^  =  (a  —  by  —  4:z 

=  (a-by-4:[(a-by-t]  =  it~S(a-by) 

OT(a-by-4:[(a-by-j-t]  =  -4:t-3(a-by} 


-by 


say. 


(1) 


and^^-i[c  +  («-^)1. 


H 


ence,  x  is 


r  IS 


expressed  in  terms  of  a,  b,  and  r 
expressed  in  terms  of  a,  5,  and  ^ 


')  ^) 


^  is  expressed  in  terms  of  a,  b,  and  c ; 
and  the  expressions  for  r  and  t  are  cases  of  (A). 

4.    (rt  -  x)  {b  -f  .r)*  +  (a  -  xy  (b  +  a-)  -  ab  (a'  +  i'). 
Let  a  —  X  -^n  -  -  2;  and  b  -}-  x  =  n  -\-  z. 


(2) 

1 

(3) 

1               ^« 

1        '^" 

1  i        ^n 

|J        act 

■ 

1        the 

1        wh 

ho\ 

red 

(1) 

whi 

\ 

{ 


\ 


%—x 


)\x-hy 


hy-]  =  t\  say. 


6-r),       (1) 


\-hy\       (2) 
(3) 


r; 

t\ 

of  {A). 


(1) 


5. 


QUADRATICS. 


233 


The  equation  reduces  to 

(n'-~  z')  [(n  4-  zf  +  (n  -  zf]  =  ab  (a'  +  b'). 
.'.  (n'  -  z')  (2n'  +  Qriz')  =  ab  (a'  +  b'). 
.'.  {n'  -  z')  (n'  +  Sz')  =  ab  (a'  -  ab -^  b'). 
z^  may  now  be  found  by  (C),  and  from  (1) 

X  =  ^(a—b)  -}-  z, 

Sz'^^ia-by  or  {(lOab-a'-b'). 
.".  a;  =  0,  or  a  —  b,  or  \(a  —  b)-\-^^(30ab  —  Sa'^ 


Sb'). 


o      2  ' 


a: 


.2 


5a;^-12  =  0. 


Find  the  rational  linear  factors  of  the  left-hand  member 

by  the  method  of  §  27,  page  116. 
.-.  (x  -2)(x  +  2)  (x*  +  2a;^  +  3)  =  0. 
.•.a:-2  =  0,  ora;  +  2  =  0,  or  a;'  + 2.V' +  3  =  0. 

The  last  of  these  equations  may  be  solved  as  a  quad- 
ratic, giving 

.T^  =  -l±:2V-2. 
.*.  X   =±1  rfc  V~~  2. 

.•.a;i  =  2;  3^2  =  — 2;     a:3=l  +  V~2;    .r^^l  — y'— 2; 
x,  =  -l  +  -^y-2;  x,=  -l--y/     2. 

Note.  In  solving  numerical  equations  of  the  higher  orders,  the 
rational  linear  factors  should  always  he  found  and  separated^  as  dis- 
junctive equations,  before  other  methods  of  reduction  are  applied. 
Such  separation  may  always  be  effected  by  the  metliods  of  §§  2G-2H, 
and,  unless  it  is  done,  the  application  of  the  higher  methods  may 
actually  fail.     Thus,  if  it  be  attempted  to  solve  as  a  cubic  the  equa- 

^'^^^  r»- 9.x- -10  =  0, 

the  result  is  .t  =  (5  +  y/-  2)i  +  (5  -  y/~  2)i, 

which  can  be  reduced  only  by  trial.  The  left-hand  member  can, 
however,  be  easily  factored  by  the  method  of  |  27,  and  the  equation 
reduces  to  {x  +  2){x^  -  2x  -  5)  ^  0. 

which  gives  .t;  =  2  or  1  ±  y/G. 


i*" 


/ 


234 


QUADRATICS. 


\ 


6.   {x-ily  ■   ./•■  ;  i2"     0. 

Factor  (see  Exam.  20,  page  113),  rejecting  constant  fa  S 


tors. 


.■.x(x-2)(:v'-2.fi-4y  =  0. 

.-.  X  =  0,  or  .r  —  2-^0,  or  :r  -  2a;  -|-  4  --  0. 

The  last  equation  gives  x  —  1  zb  ■/-  3. 


V 


i 


Ex.  61. 

Solve*  the  following  equations : 

1.  (x  +  a-i-hy  =  x^-\-a'  +  ly\ 

2.  {x-{-ai-hf-^x^-i-a:'-{'b\ 

3 .  (a-h)  x'  4  -  (b  -  x)  ci'  +  (x  -  a)  P  =  0. 

4 .  (rt   -  b)  x'  f  (.r  -  b)  a'  -i-(x  +  a)  b''  :=-.  2  abx.      . 

5.  (x a)'-\-(a~  bf-{-(b-xy^O. 

6.  (x~ay-\-(a-byi-(b-xy  =  0. 

7 .  (a-^  -  b)  x'  +  (.T^  -  a)  b*  +  (Z.'^  -  x)  a*  =  a5.r  (a^Z* V  -  1 ) . 

8.  (x  -  a)  {x  -  Zj)  (a  -  Z^)  +  (x  -  Z*)  (.^•  -  c)  (b  -  -  e) 

+  (x  —  6')  (^'  —  a)  (c  -  a)  =  0. 


9. 


10. 


X" 


x~  1 

x''-l 
x*~l 


=  0. 


0. 


11. 


12. 


X 


16 


X 


1 


=  0. 


x''-l 


X 


,4 


0. 


13.  x'-\-5x^--lGx''  +  20x-~l(j  =  0.     (See  §  21.) 

14.  X*  ~  3x'  +  5:r  +  Gx  +  4  --  0. 

15.  (x-ay  +  x'  +  a*-^0.       17.    .r  (a; -- 2)'^ (.r  +  2)  =- 2. 

16.  2a^  =  (x-Gy.  18.    (4.T^--17):i'+12=:0. 
19.  x'  +  (t/7*  !- 1)'^  -  (a'  +  b')  {x'  +  1)  +  2(a'^  -  b')  x  4-  1. 


/ 


} ' 


constant  ki  \ 


X. 


'(a'h 


b  -  -  c 


H^ 


--I). 


21.) 


\x 


\x 


+  2)  =  2. 
+  12  =  0. 
//)  X  -f  1. 


QUADRATICS. 


235 


').    .^■^.^•-lG9)■H17.^•--^^.^•^    -3540. 
^1.    6.7:(.r  +  iy^  +  (2a<-'^-[-5>'---150.T+l. 
)     12.    2.f(:r-l)2  +  2--(.r+l/.     24.    5.r*  -  12.r^  +  1- 


/     ^3.    .f'  =  12a;  +  5. 


26.    «-f--A_+^_    +_iL_ 
.r      :i'  —  1      .r  —  2      .r  -  3 


/ 


27. 


(^•+iy 


7y/- 


25.    (.r  +  4j^^3(2.r    -  1)'^ 

.r  —  4      .r  —  5 
(a;^+l)(^-^+l)_m 


) 


{x'-\-l)ix~-\f       h 


'    28.    i^^L^'I^. 
I  x{p^  ■\-  1)       n 

■      29     (.'^M-l)G:rM-_l)_m 
(.'^ -f  l)(a:*+l)       n 


30. 

(.r'^-l)(r'-l)       vi 

31.    G^^'fl)G^''^+l)^m 


32. 


a;''(.r+l) 


Wi 


33. 


34. 


35. 


36. 


37. 


38. 


x{x^Vf 


x{x'-\-\){x-\f      n 
n  (Vi  —  7)1) 


{x'  +  1)  {x  -  IJ'      2m. {2m  -  n) 
{.v'+iy  _    4m2 


7W  —  n 


x(x'-iy 

(x-l){x'-}-iy  ^  2(m-ny 
{x^  ~l)(x-\-  ly  mn 


x" 


2m 


(.r+  l){a^-\)      2m- n 
{x''~\){x-{-iy  _on.-\-n 


n 


{x'-{-\){x-\y      m 

(x^  1)(a-*  +  1)  _  m-\-  n 
\x  —  I)  (.1'*  —  ] )      7/i  --  n 


39.    .r^ 


40.    .!•' 


ax  — 
bx  — 


a 


ax 


bx  — 


~b 


41, 


42. 


ax 


-b 


bx  ~ 


a 


X    -^ 


a 


ax^  -{-  bx  -[-  c 
a  +  bx  -\-  ex'' 


/ 


236 


QUADRATICS. 


43.  x'^ix-iyix'  +  l). 

44.  aV  =  (a  -  a;)Xa' -  37^). 

45.  x'  =  (x-ay(x'-l). 

46.  m(x-{-'m  —  n)(x  —  'm-\-7ny 

=  n(x  ~  m  -{-  n)  (x  -\-  7  m  —  rCf, 

47.  7n?{x-\-m-\-\ln){x  —  7n~bny 

^=n^  {x-\-  Vl7n-{-n)(x~  5m  +  7^)^ 

48.  m'^{x-{-rti-\-VJ7i){x  —  7n-{-lny 

=  7}?  {x  -\-Vl  m  -{•  n)  {x  -{-*1  m  —  ny. 


49. 


50. 


a     a; m 

X     a      n 


x^-j~ax-\-  a' 
x^  —  ax-\-o? 


-„     a~ X  .  x~  h 
5d.    r-J 

a;  —  6      a  ~  a; 


51. 
52. 


ar'^  +  a'* 


x^  —  ax-\-d^ 
x'^a? 


=  c. 


{x-^aj 


~c. 


54     2aM-a(a-a:)  +  (a  +  a:y  ^  g+1. 
2d^-\~a{a-{-x)-\-{a  —  xy      c-~l 

55.  a;*  +  (a  —  a;)*  =  <?. 

56.  a;*  +  (a;  -  4)*  =  82. 

57.  {a-xy-{-{x-hy  =  c. 

58.  a;^  +  (a  — a:)^  =  a^;   a;^  +  (6  -  a:^  =  1056. 

59.  (a  -  a;)^(a7  -  hy  +  (a  -  a:^  (a;  -  hy  =  a^^>^(a  -  h). 

60.  (a;-a  +  5)'-(a;-a)'  +  (a7-5)'-a:'  +  a-''-(a-^>) 

=  {a—l^c^. 

^^     {a-xy  +  {x-hy_a'-\-h' 


\ 


-~¥ 


62. 


{a  —  xy  +  (a;  -  ^>)'      a^  +  i"^ 

(g  —  xy  +  (a;  —  ^>)4  ^  a*  +  ^>* 
(«  -  a:)^  +  (a;  -  i)'      a'  -  i^* 


/ 


A 


(a -5). 


^.: 


I    f 


H 

!    / 
1 


73. 


74. 


75. 


76. 


77, 


QUADRATICS. 


63. 
64. 
65. 
66. 
67. 
68. 
69. 
70. 
71. 


(a  —  xf  +  (x  —  bf  ^  (t'  -  -  b"^ 
(a  -  xf  -{-(x-b) 


3         f^3  _  ^3 


(a_^)3     ^     (^-^)3_a3     ^     ^3        ^ 

6  —  a;  a  —  a;        6 


a  —  X 


+ 


a  —  a;        a       a 
X—  b         a      b 


a 


{x-by      {a-xf      b' 

(a-xy-{-(x-by_  a'-\-b' 
{a-^b-2xy  {a  +  by 


a'-b'' 


(a-xy-{-{x-by , 

{a-\-b-^xy  {a-\-by 

(g  -  xy  +  (or  -  by  _  .       ^y 

(^a-xy  +  {x-by    ^      ^' 

(a-xy-(x-by__       (a-b)c 


{a  —  x)  —  {x  —  b)       (a  —  x)(x  —  b) 

(a  —  xY^-ix—by        /  N  /        7  X 

(a  —  xy  -\-{x  —  by 

(a-xy-^(x-by_        c 

{a  -  xy  ■\-{x-  by      (a  -x)(x-  b) 


72.  {v-{-x''y  =  {x^-zy 


x'-\-\      _a 


2x(x^-\-l)      b 


78. 


{x-{-iy  ^a 


X 


{x^  +  1)       b 


{x^-iyix'-\-l)   _a    _     x{x-\-\y_ 


{x-iy{x''-X'^\)    b 


(:^-iy 


X 


a 


( 


X' 


X 


-{-ly 


{x-{-iy{x^-\-\)      b 

jx'  +  1)^  _  g 
X  {x  4-  ly     b 


80. 


81. 


82. 


a 


(.r  -  1)' 


23  ■ 


r.r'+a.-+lT.r' 

L(.r+iy^io 


-x- 


a 


x-\f 


X 


x''-\-l 


a 


ix'  - 


\y 


x{x^  -\-  l)_g 
{x'   - 1)^  ~  b 


i 


1* 


'•I 


238 


SIMULTANEOUS    QUADRATICS. 


'    (.'•-l)(.r'   -1) 
(.;«-l)(.x-^+l) 


(I 

-  • 


85.    ^^'-±1^  =  ^. 
.,'  +  1         b 


86.    (:^i^ 
1 


.1-^ 


a 


§  49.    Quadratic  Equations  Involving  Two  or  Moi 

Variables. 


1.  (x  +  y)(x'  +  f)  =  a, 

(1)4-2(2),   .•.(.T  +  y)^.=  a  +  2^. 
:,x^y  =  ^{a-\-2c). 
(Any  one  of  the  three  cube  roots.) 


(•'5) 


(1)  +  (2), 
By  (3), 
Also, 


^■'  +  y' 

xy 
X  -  y 


'x-y\_ 


a 


2c 


</{a-\-2c) 
+  y   -V(^  +  2.) 


r/  +  2e 


^•^        -^(a4-2c) 
^_V(a  +  2c)  +  V(a-2g) 


D. 


2^(a  +  2c) 

,.-V(^  +  2c)-V(a-2c) 
^  2^(a  +  2c) 

Not  any  one  of  the  six  sixth-roots  of  a-\-2c  may  be  used 
indifferently  in  the  denominator,  but  only  any  cube-root 
of  whichever  square  root  of  a  +  2(?  is  used  in  the  numer- 
ator. Thus,  if  the  radical  sign  be  restricted  to  denote 
merely  the  arithmetical  root,  if  k  be  defined  by  the  equa- 
tion F  —  Z:4- 1  =  0,  and  if  m  and  n  indicate  any  integers 
whatever,  equal  or  unequal,  the  value  of  x  may  be 
written, 


Si 


i    "V^ 


TICS. 


SIMULTANEOUS   UTADRATICS. 


2:19 


\ 


c- 


1 


.1-* 


•2? 
f    2 


[NO  Two  OR  Mop! 


(       ^''trst  Met 


1/ 


^  /  ,. 


£}. 


11  o;'^  -  82'y  +  bif  -  13 (.r  +  y),'  (2) 

///Of^.     Eliminate  (•^'  +  y)- 

.  • .  1 04 .<;"^  -  65  xy  +  39  y^  -  -  99  .r'-'      72  xi/  +  45  /. 
.■.bx''\-1xy~(jif  =  0. 
:.{bx-^){x-\-2y)  =  \). 
.-.:/;--  f?/  or  ~22/. 

)  Substitute  these  values  for  x  in  (1), 

.-.  72.7/  =  3G0?/,  or  45?/  ~^  -  9?/. 

.-.T/^O,  or  5,  or  -\, 

and  a:  =  0,  or  3,  or  f . 

^^econd  Method.  Take  the  sum  of  the  products  of  (1)  and 
(2)  by  arbitrary  multipliers  h  and  I. 

h^^x"  -  5.iy  +  3/)  +  /(11a:'  -  ^xy  -[-  5?/) 
=  (9X^+130G^'  +  y).  '(3) 

Determine  Z;  and  I  so  that  the  left-hand  member  of  (3)  may, 
like  the  right-hand  member,  be  a  multiple  of  x-^y.  This 
may  be  done  by  putting  a;  =  — ?/  in  (3),  from  which 

167j  +  24/  =  0. 


<-i  ic 


SI. 


2  c  may  be  used 
dy  any  cube-root 


in 


the 


numer- 

:ricted  to  denote 

Ined  by  the  equa- 

3ate  any  integers 

of  X   may   be 


.•.if;^-=3,  l  =  ~2. 

Substituting  these  values  in  (3),  it  becomes 
2x'^  -\-xy  —  y'^  =-■  x-\-y. 
.-.  (re  -f  ?/)  (2x  -  y)  =  a;  +  y, 
or  (x  -\-  y)  (2a;  —  ?/  —  1)  -=  0. 

.•.  either  x  -{-y  —  O,  or  2x  -  y      1  —  0. 


y 


-  X,  or  zx 


2. 


1. 


240 


SIMULTANEOUS   QUADRATICS. 


Substituting  these  values  for  x  in  (1),  it  becomes 


16a;'  =  0,  or  10a;'  -  7.r  +  3  --  27a:  ~  9. 
a:  =  0,  or  8,  or  -J, 


\ 


and 


y 


0.  or  5.  or 


x^  +  if  ^  a^  U" 
ar*  +  ^Z* "~  a^  +T«' 

x"^  -\-  xy -\- y^  =  o?  -\- ah  •{■  Z»'. 


(1)^(2), 


X' 


-3^y-\-  xY  —  ^y  +  y 


(a;'  +  y'f  +  xhf 

^  a'  -  a^b  +  a'b'  -  ah^  -\-  b* 
(a'  4-  bj  -  a'b' 

a^y  -{-  xf       __       d^b  -}-  ab^ 


(ar'  +  fy  -  xY      (a'  +  ^'0'  -  «'*' 


Write  z  for        ..  ^  „.  and  ^  for 
(3), 


aS 


X'  +  y'' 

z 


d'  +  Z>^ 


1-2 


l-;5;' 


/;  or  —  -• 
A; 


xy     _    «5 


or 


—  a5 


a:y         _         ct5 


or 


a'  +  ^'' 


o^-\-xy-\-  y^      c^-\-ab-\-  b^        a'  —  ah  +  i' 


(2). 


xy  =  ao  or  (a^  +  6  )  — ' —    '      « 


(4) 


V[(2)  +  (4)]. 

:.x-\-y^±i{a-\-b), 


or 


V(2a'-a^>  +  2Z»')^/^ 


-  a6  +  b'' 


iTICS. 


it  becomes 
f  3=rr27:r 


h\ 


y  ~  a'b' 


SIMI'LTANKOrs    Ql' APRATIfS. 


(3)   i  4- 


241 


3(4)], 

x-i/-±(a-b). 

,v(2«»+«*+2*-)^«:+j+j:. 

• 

.*•  X  ~±a,±  h, 

i[V(2a'-ai  +  ^') 

+'^('-"'+'^''+'^''sy:lti- 

y-±zb,±a, 

^[y/CZa'-ab-i-b') 

,v(2.'+«i+2*o]^«+:^j. 

(.r'  +  if)  (x'  +  7/)  -  a, 

(1) 

{x-\-y){x'-\'y')-b. 

(2) 

or 

a'  +  b' 

a' 

~ab-\- 

-hb' 

\+b' 


(4) 


Put 


(1)' 


*»  

xy 

z  — 

1- 

a:'  +  y' 

-2       a 

l-2z'     b 

2az^-bz-{a-b)  =  0. 

4:az  =  b±  -y/iSa^  -  8ab -{- b')  ^  b -\-r,  say. 


x]/     _  ^  + 


^'  +  y 


a 


(3) 


a: 


±^=-.^     1^^-  +  ^  + 


^-y 


J 


a 


X  _  V(2a  +  ^>  4-  r)  +  V(2a  -b-r) 
y      V(2a  +  ^>  +  r)  -  V(2a  -b-r) 

^  rV(2«  +  ^^  +  ^O  +  V(2a  -b-  r)Y 
2{b  +  r) 


(4) 


(a;^+y^  +  2a;2/)(a;HyT[(^'+2/')-^yr  =  «' 


(3). 


_«lMCI.X.N-,,„rs  „,,,„,^,^,,.^ 


^■'^L  (TTTv^ ~ha' 


(i). 


il^-j-j-y 


in  which      r^^   /fo   2      o    ,  ^J 

-tiio  value  of  ?/in.,v  K^  i    •      i  1 

fb™  in  (4/  "'"^  ''^  ^'---'  from  tl,at  of  .  ,y  ,,,  J 


U  1 


li 

I 

ii 


fit  y,4  __  , 

^  —ctx  —  hy^ 

X  X  (1)  -  -  y  X  (2), 

.*.  either  ;r  —  ?/  =  o 

•  in  which      t -    /r/^  ,   ,J -^  ^       *^^(^  +  :^). 


(1) ! 

(2)    i 


(3) 

(4) 

(5) 

(6) 

(7) 

(•8) 
(9) 


6. 


Ls 


(1) 


Ji.VTirs. 


-.-.-_A  >  - 


SlMrLTANKOrs   (irADRATH'S. 


w 


rj[    h-ir    ) 

r)(4a  -b~~  rf 
\a-~b~rf]  ' 

lat  of  X  by  tlio  first^ 


\ 


[0)  +  («)]. 


*2^ 

f    2 


rO  :  (10)    .-..i.'-l-y        ,     ,,       , 
;  11)  -.  (10)', 

/    )  ^     2a-^ 

-</(26  +  0" 
V(2a     0 
^(2/H-/)" 


sy(2^»  +  0' 


.r   -y 


(1) 
(2) 


<  (10)  and  .^•  h  y  =  V4I^^- 


and 


,^-V(2^-hO  +  V(2a--0 

__  V(2/>  +  0~V(2rr,      Q 
^  ^(26  +  0 


6. 


/)■ 

+  y). 


y), 


(3) 
(4) 
(5) 
(6) 
(7) 

(8) 
(9) 


Let 


in  which      i  -=  VK  +  2 «&  +  5  Z*'). 
a;*  _  c*  =  m  (a;  +  ?/)*, 


^-2/ 


-^  +  1 


2x 


and  2  —  1  = 


2.'/ 


(l)  +  (2), 


X  —  y  ^      y 

x^  -\-  if  =  rti{x  -\-  ?/)*  +  n  {x  —  yf, 
(2+iy  +  (2-l)*=16(7?iz*  +  n 


)■ 


(8m  -  1)2*  -  62'^  +  (8n  -  1)  =  0. 


2l:i 


(10) 
(li.) 


(1) 

(2) 


(3) 


m 


244 


.SIMULTANEOUS    QUADRATICS. 


j3-hVr9~(8m-l)(8n-l)l         ,, 
\  8m -1  ^  ^ 


(2)  and  (3),     (z  -  iy(x  -  y)*  +  16  c*  =  16n(a;  -  y)*.        j 

2c 


x-y 


</[16n-(.-iy] 


an( 


^  +  y  =  -T 


1C2; 


f5) 
1 


:r  = 


-^[16n-(z-l/] 


and 


c{z-V) 
and  the  value  of  z  is  given  by  (4). 


7. 


^  +  y*  =  mw. 
•••  (a;  +  y)' -  2a;y  =  K2m  +  n'), 
and  (a;  +  yy  —  3  .ry  (^  +  y)  =  mn. 

Let  w  =  a:  -}-  y  ^^^  "^  — '  ^3/'  ^^^^  ^^^  equations  become 
u'-2v  =  i(2m  +  n''); 

u^  —  3  wy  =  ?/m. 
Eliminate  v, 

u^  —  (2  m  +  w''^)  II -\- 2  mn  =  0. 
w*  —  (2m  +  ^i'O  ^^  +  2mnu  -^--  0. 

i/,2  _  97i  =  -4-  (7i^^  —  772,). 

(the  value  w=0  was  introduced  by  the  multiplication  by  w), 
or  it^  -{-nu  —  2m  —  0. 


9. 


I 


SIMULTANEOUS   QUADRATICS. 


245 


m  (t) 


y)'-     1 


(5) 


L+D- 
1)<-16otz'' 


V 


( 


?r 


?;i 


). 


or 


Also. 


it  and  i»  are  completely  determined. 
X  -\-y  -^  u,  X  —  y  —  VC*^^  ~  ^^)- 


a:  ~^\u 


[^c-\-^{u'~4.v)] 


•••y==ir^^-V('^'--4v)]. 

If  ?;i  —  7  and  ii  =  5,  the  above  equations  become 
cc^-^i/=  13,  and  a;'  +  y*'  =  35. 

Solving,  as  above,  gives 

w  =  5,  or  2,  or  —  7  ; 

2i;  =  12,  or -9,  or  36. 
.'.  X  -\-  y  =  6,  or  2,  or  —  7  ; 

^  -  y  =.  -t  1,  or  =fc  V22,  or  ±  i  V23. 
.-.  :r  -  3,  2,  K2  i:  V22),  or  i  (-  -7±:^  V23)  ; 

y  =  2.  3,  J(2^=V22),  or  H-7^^■  V^S). 


)ecome 


m' 


•^  +  2/'  =  f  • 

Testing  this  for  rational  linear  factors,  it  is  easily  reduced  to 
(y-iy(f  +  2y  +  i)^0. 

.•.y=l,  ori(-2±:V2); 
^  =  i,ori(-l=h4V2). 


cation  by  u), 


9.  (2a:-y  +  2)(.r4-y  +  2)  =  0, 

(x+2y-  z)  {x-\-y-\-z)^  1, 
(a:  +  ?/-22)(a;  +  y  +  2)  =  4. 

Let  s=^ x -\-y -\-z,  and  the  equations  may  be  written 


(1) 

(3) 


246 


SIMULTANEOUS   QUADRATICS. 


(s  +  7/  —  2z)s=l, 
(s-3z)s  =  4. 
(4)  +  3(5),       (4s  +  ^  +  2/-62).9-12, 

or  (5s  -  7 z)s  =  12. 

3 (7) -7 (6),    [(15s-2l2)-(7.9^-2l2)].9  =  8. 

.-.  8s'^  =  8.     .\s  =  ±l. 
Substituting  in  (1),  (2),  and  (3),  they  become 
2x--'i/-\- z  =  ±9,  x-\-2y —  z  = 
x-\-y—2z  =  ±4:. 
.*.  a;  =  ±  4,  y  =  =F  2,  2  =  =F  1. 


1. 


(4) 
(5) 
(f>) 

(') 


10.  x^-j-7/^  —  a, 

xy  -}-uv  =  c, 
xu-\-yv  =  e. 
Let  i^=xy  —  uv. 

(x  ~yy  =  a  —  c  —  t. 

y=^W(a  +  c  +  t)--,/(a--c-t)l 

{u  —  vf  —  h  —  c-\-t. 

U=l[y/(b  +  C-t)-{--^{h-C-^t)\ 

Also,  2  {xv + yv)  —  {x-\-y){u-\-v)  +  {x—y)(u—v)  —  2e. 

J^^[{a-c-t){h~c-Vt)]  =  2c. 

.-.  [4c''-^{a~-c-t){h-c~\-t)-~{a^c-\-t){h-{-c-i)J 
=  \Qe' {a-  c  —  t){h  -  c  -\- 1), 


U 


Le 


AL 


Als. 


Elin 


(7)  ai 


(4) 

(5)  ^\ 

(6)  \ 

0) 


il. 


0]. 
0]. 


|f  01- 


2  c. 
■t)(b+c-i)J 


SIMULTANEOUS    QUADRATICS. 


247 


[(a  -  by  +  4:e']e  -2(a'-  b')ct 

+  (ai-byc^-4:e\ab-\-c^)  +  4:c*  =  0. 

._(a'-b')c±:2c^l(ab-c')\(a-by-^(c'-e')] 


{a-by-\-4:c' 


11, 


XT/ 


tlV. 


x-^1/-{-u-\-v  =  a, 
a^i-7/-{-ic^  +  v^  =  ¥ 
o(^ -\- y' -\- u^ -\- v^  =  c" 


Let  x-\-y=  2  (rt  -f  z). 


u 


-{-v  =  h{a  —  z). 


Also,  le^•.  7'-=xy  = 


uv. 


(1) 

(2) 
(3) 
(4) 


(5) 
(6) 


(a^  +  3/)'  =  ^  +  /  +  3a;y  (:^  +  y), 
(w  4-  'y)^  =  t^^  +  'y^  +  3  z^v  {u  +  ^;). 


a 


Also, 


0) 

{x + ?/)'  =  x'^-t  ?/' + 5.i;?/  (a;'+  ?/')  + 1  OA-'y'(:?; + y) , 
(u-}-vy  =  u'+  v' + 5  wv  (  w'4-  -y')  + 1 0?^'i;2  (z^ + v) . 
a  (53*+10aV+  a*)  =  16(c^+ SiV+lOa?-^).   (8) 


Eliminating  r  between  (7)  and  (8), 


313 


45aV  -  30« (a'  +  2b') z"  +  a'  -  20a'b 
-80//+144ac5  =  0. 

15as^-5(a^  +  25') 

=  ±  2 V[5  (a^  +  5 bj  -  180ac^]. 


(0) 


\  3a 


(7)  and  (9),     12ar  =  a'-4:¥-{-S 


az' 


2a'-2P±.2y/ll[(ia'+5by-SQac']\ 


_b(a'~b')±^l5(a'-\-^by-lS0ac'l 


SO  a 


(11) 


248 


SIMUI/l'ANEOr.S    QUADRATICS. 


(10)  and  (11)  give  the  values  of  z  and  r,  which  may  now  be 
treated  as  known  in  (5)  and  (6). 

a;  4-  ?/  =  I  (a  +  2),  and  xy  =  r. 
.'.x-i/^hy/[(a  +  zf~16r]. 
.'.x=:\\a  +  z±^[(a  +  zy-lQr]l 
i/=]\ai  z^^[(a  +  zy-ie,r]\. 

The  vahies  of  u  and  v  may  be  obtained  from  those  of  a;  and 
y,  respr  -;tively,  by  changing  z  into  —z. 


12. 

(1)  -H  xyz, 


ax 

a 
yz 


bu'~  cz  —  --\ 1 — 


X 


zx     xy 


y 


xy  -{-yz  +  zx 
Also  from  {l)--r-x7jz, 

yz      xyz\x     y      zj 


xy  +  yz  -\-  '''PC 


,2«2 


x'y'z 


(2)  X  (3), 


7fz^ 


xhfz^ 


:.  a^x^  —  a-\-  h-\-c. 


13. 

(1), 
then 


y-\-z  —  x  _  z-]-x  —  y  ^  x-\-y-~z^ 
a        ~        b  c 

xyz^=w^. 
z  X 


y     m 

a-\-h      h-\-c      c-\-a      r 


,  suppose ; 


xyz 


{a^b){b-\-c){c-\-a)      r 
...r^  =  (a  +  i)(6  +  c)(c4a)- 
Hence  the  value  of  r  is  known,  and  from  (3) 
rx  =^  m{b -\- c). 


(1) 


(2) 


(3) 


(1) 

(2) 
(3) 


(1)1= 


may  now  hi 


tiose  of  a;  and 

(1) 


-|-  '/.X 


(2) 


(3) 


ppose 


(1) 

(2) 
(3) 


SIMULTANEOUS   QUADRATIOS. 


249 


14. 


z  -\-x  =  2  hxyZy 
X  -\-y--~2  rxyz. 

^       y  -  f  ^  -^  -  +_^*  =  ^  +  V  =  ^  +  .y  +  ^ 

^'^        2a  2b  2c        a-\-h^c 

X 


(1) 

(3) 


?/  z 

b-\-  c  —  a      c  -\-  a-\-  b      o,-\-b  —  c     ^  ■' 


.'.  a^y\^  — 


xyz 


{b-{-c~  a){c  -\-a  —  b){(t  -\-b~c) 

'^  (b-{-c-a){c  +  a  —  b)(a-\-b~cj 

Hence  the  value  oi  x^y^'^  is  known,  call  it  -,  and  substitute 


in  (4) 


1_         X 

T      b  -\-  c  —  a 

rx  =  b  -\-  c  —  a, 


15, 


in  which      r'^  =  {b  -\-  c  —  a){c  -\-a  —  b)(a  ~{-b  —  c). 
y'^  +  z''  —x(y  +  z)  =a, 


z"^  -[- x"^  —  y  (z  -\~  x)  =  b, 
x''^y''  —  z{x-\-y)  =  c. 


(1) 
(2) 
(3) 


(1)  +  (2)  +  (3), 


{pc^ '\- y"^ -\- z^  —  xy ~ yz  - zx)  =  a-\-b-\'C.    (4) 


(1)  may  be  written 


t 

i 


^'  +  2/'  4-  2'  —  x(^x  -I-  ?/  4-  2)  = 


a. 


(2)  may  be  written 


X- 


-\-y'''\-z'~-y{X'\-y-\'z)  =  b. 


(3)  mcy  be  written 


X- 


H-  y'  +  2'  -  -  ^  {^'  +  y  -f  2)  =  c. 


(5) 


(6) 


0) 


250 


SIMULTANEOUS    QUADRATICS. 


a 


b      h 


a 


y~x     z  —  y     x~  z 

a'^  -f-  u^  -f-  &  —  ah  —  he  —  ca 

^'^  +  y'^  +  2:"''  —  0:3/  —  yz  —  zx 

_2(a^~{-h'^~\-c^—ah—hc—ca)        ^q^ 

— r7"T ^^ 

a-f-  o  -f-  c 

^  2(a'  +  h'-\-c^-3ahc) 
(a  +  b  +  cy 

Write  r'  for  2(a' i-P  +  c"" -S ahc). 

(9)  ...^  +  y  +  ^  ^ 


(4) 


(9) 


a-\-h  -\-  c 


Returning  to  (8), 


(4) 


/     ,       ,    xo     2(a}-\-h^-\-c^—ah—hc—ca) 

a+6  +  c 

a-{-h-\-c 


l[{^)^{ll)\x^-\-f^z^ 


d'  +  h^^c' 
a-\-h  -\-  c 


(10) 

(8) 

(11) 

(12) 


(5)  and  (10),  X'  +  f  +  z- -     ,7,     =  a. 

a-\-  0  -f-  c 

.'.  rx  =  {a-i-h-\-c) {x^ -\-y'^-\- z^)  —  a{a-\-h-{- c) 
(12)  .  =d'-\-h''  +  c'-o.(a-{-h-\-c) 

-=h''-{-c'-a{h-\-c). 

(5),  (6),  and  (7)  are  symmetrical  with  respect  to  (xyz\ahc) ; 
(10)  shows  this  substitution  does  not  affect  r,  and  conse- 
quently the  values  of  y  and  z  may  be  written  down  at 
once  from  that  of  x. 


SIMULTANEOUS    QUADRATICS. 


251 


;ten  down  at 


Ex.  62. 

1.  G[(7-.r)H2/']-13(7-:r)3/;   ar^  +  4y  =  3/H4. 

2.  10x'-97/-^2x^;    8a;'- 6?/^  -  13.r. 

3.  xi/  =  (S-xy=:(2-y)\ 

4.  x'-{-y''  =  8x-\-9y=^Ui. 

5.  .r'4-2/'  =  ^  +  y  +  12;   rry  +  8-2Cr  +  3/). 

6.  a;  -{-  ^2/  +  2/  =  5  ;   a;'^  -f-  a;^/  +  y''*  —  7. 

7.  x''-i-7/=7xi/--=28(x-{-i/). 

8.  0;^^  + a;?/ +  3/'-- -—-!==  — 

X  -f-  y       xy 

9.  ^-^  4- a:y  +  ?/*  =  133  ;    a;^?/  +  a;y  +  V  =  114. 

10.  (a;  +  y)(:i-2  +  ?/0  =  l'7.ry;  {;x-y){x' -y')^9xy. 

11.  25(.r=^  +  3/^)  =  7(a;  +  2//  =  175a:y. 

12.  2a;'  -  2/'  =  14  (a;'  -  23/')  =  14  (.r  -  y). 

13.  2a;'-3a;y  =  9(a;-3y);  3(a;'-3y')  =  2(2a;'-3a;2/). 

14.  2a;'-a:y+5?/'=10(a;+y);  a;'+4a;y+3/ =- 14(0;+?/). 

15.  (2a;-37/)(3a;  +  42/)=:39(a;-2?/); 
(3 X  +  2 ?/)  (4  x  -32/)  =  99  (.r  -  2?/). 

16.  {x-\-2y){x^Zy)  =  3(a;+2/)  ;    ( 2:^+7/) (3a;+2/)  =  28(a;+2/). 

17.  a;  +  y  =  8;    a;*  +  y*  =  706. 

18.  a;  +  2/  =  5;    a;^  +  2/^ --- 275. 

19.  a;  +  y  =  2;    13(.r^  +  y^) -- 121(0;^  +  ^'). 

20.  a;4-2/  =  4;    41(a;^  +  ?/)  =  122(a--*  +  :/)• 

21.  x^  —  5a;2/  +  2/^  +  5  —  0  ;    xy  =  x  j-  ?/  --  1. 

22.  a;-'  +  2/  =-  5  (.c  —  2/) '-    -f  +  2/  =  2  (.6-  —  y). 


252 


SIMULTANEOUS   QUADRATICS. 


23. 
24. 
25. 
26. 

27. 
28. 

29. 
30. 

31. 
32. 
33. 
34. 
35. 
36. 
37. 


10  {x^  -f-  3/)  -  10  {X  +  if)  -  13  {x'  +  f). 

:c-.L-a:y  4-y  =- 5;    a:^  +  a:y +  y^  =  17. 
X  -!    v  -:  '^ ;    (a;  +  1)^  +  (y  -  2y  -  211. 

3(...-lX.+l;.4(.+l)(,-l);g^;^|g^; 

,       1 

xy 

.r  +  y4-l--=-0;    :r'' +  ?/ +  2 -=  0. 
:i--|-?/=l;   3(.r«  +  3/«)  =  7. 

•!.r//     -r,(r,     -.r);    2  (.^''^  +  ?/)  =  5. 

127. ry       17;    9 (.^;^  +  t/^)  ^  -  8. 

(./-  !-///  l--4a;y  =  5-12?/;   y  (^r^  +  ^/^^  _|_  3  ^  q. 

./;  h  y  --^  -^y ;  ^'  +  y'  =  o:^  +  ?/. 

.<y^+3y-l)  -  2y^+2y+3  ;  y(a:^+3:r  -1)  -  2:i^*^+2:i'+3. 
x^  ,  ?/ 


ci'      //  'a      Z>         \a      6, 


38. 

39. 

40.    X  i-y  —  a; 


x^  +  xy"^  =  a;   ^ -{■  x^y  =  b 

X 


] *^  =  c. 


41. 

42. 
43. 
44. 


h-y 

x^  -f  ^if  =  ^  '      ;    rf-f-y^  —  (a''  — l)y. 
.-c  +  y^  =  a-T ;    :r'^  +  2/  =  ^y- 

a:  +  y'^  =  (^y^ ;  •'t''*  +  y  ~  ^>'^'^. 

^*   -  y *  ~  a'^  {x  —  yY  ;    a;"'  —  x'^y  -|-  ajy'^  —  y^  =  i''^  (a:  -f  y ) . 


SIMUT.TANK(3rS   QUADRATTrS. 


253 


39W-:/+i> 


+  3  =  0. 


2:i^'^+2a:+3. 


\h'(x-{-y). 


45.  (x  +  y)(x'-{-Sf)^-m-    (x^-y){x' -\-^if)^-:  n. 

46 .  xhf  —  'i/(a  —  xf  ~  x(b  ~  yf. 

47.  x^  (J)  —  y)  = 'if  (a —•  x)  ~  [a -—  xy(h  —  yf. 

48.  ce{x'-^e)^h\x-\-yy;    a\y'-\-^)  =  c\x-^y)\ 

49.  a;'  —  y''  =  a  {x^  —  if)  ;    x^  -\-  y^  =  h  {x  ^  y). 

50.  a:  -f-  ?/  =  a  ;   a;''  +  3/^  =  hxy. 

51 .  X -{- y  =  xy  ^=  x^  -\- y"^ . 

X 


52.    x  —  y 


y 


x^  —  y^ 


53.    ^^^(l  +  2/')(l+yO  =  «;    a:'(l  - 3/^)0 -y)  =  i. 

54.  ^''  +  ^y+y'^^'  +  r^^y. 

x'^  —  xy  -\-y^  a  h 


a 


55.  x^y^rxy'^———'   x'y-\-xy'  =  h. 

x^  ■ '-  y 

56.  x'^y-\-xy'^~a{x^-{-y'^)]   x^y —  xy^  =-h{x^  —  y'^'). 


58. 
59. 
60. 


61. 


y       X 

x^  +  ?/''  =  ax^y"^  =  a:?/  (.r  -f  y). 
aia;y  =  a  (o;^  +  ?/)  —h{x-\-  y)^. 
xy  (x  -\- y)  =  a  ;   o?y^  {x^  -\-y^)  =  h. 

Vo;       y/  \x     y) 


62.  a;*  +  y*  =  '^ {p^  +  ?/) ;  ^'^  +  ^v  +  y'^  ~  ^• 

63.  a^>(a;  +  3/)=-a:?/(a  +  ^>);    x^ -\- y"" --- a^ -^  h^ . 

64.  .r^  -}-  y3  _  ^  ^^  _|_  y-^  .    .^4  _|_  y  _  ^  (^j^  _u  ^y^ 

65.  a;''  +  ?/2  =  a;    x'-\-if=-h{x'^if). 

66.  .i-y  --=  a  ;    x^-\-y^  =  h  (x^  +  y^). 


,W' 


254 


SIMULTANEOUS   QUADRATICS. 


77.    x^  —  y^ 


a" 


a?  —  'if  —  <?' 


67.  (.'-  -?/)(.r'' -I -?/)-=(«-/>)  (a'»-l//'j;   x'-y'-ti'      //. 

68.  x'^    -  y^  —  <i ;    .r*  -\-  y^  -  Z>  (.r  -  y). 

69.  .r  I  y  -^  r^ ;    .'J;*  -  \-  y*  ^"  />. 

70.  x-\-y  —  a',   x^  +  y^ ^    b. 

71.  .r  [-  y  —  a  ;    o;'^  +  y^  --  h'^x^y'^. 

72.  .r  +  y-a-f-Z»;    (^  -  Z>7(.r* -f  y)  =.  (a:-yy(a*  + ^>'). 

73.  x\-y---a\    c(a;* +  ?/)-- a,-y(.r'' +  y'). 

74.  (.'<;-hyy~r<(a;'  +  y'0;   .ry^  ^(a;  +  y). 

75.  x'^y  +  ^^y^  —  «■' ;  ^^  (^^  +  y")  =  ^'V- 

76.  .r''  --  a  (x^  +  2/')  —  ^^y ;  2/'  --  ^  (^"^  4-  y'*)  —  «^y. 

78.  x'-7/=d'xy]   (.TH?y7  =  ^'(-^-'--2/')- 

79.  (a;  +  y)  ^V  ~ci'\    ^^  "h  y^  —  ^• 

80.  (a;  +  y)  ^2/  =  oi ;    ^'^  +  2/^  =  ^^'2/- 

81.  .T*  +  y*  =  a  (:c  4-  yY ;   ^'^  +  2/^  —  ^  (^''  +  y)^. 

82.  .T*  +  ^y  -f-  y*  =  «  ;   a;'^  —  a:y  +  y'^  =  1. 

2/(1  + ^'2/)     ^U-''^yJ 

84.  n-'^  +  y^  =  a  (.'I' +  2/)  5   a;*  +  y*  =  Z>  (.x^  +  y^). 

85.  .r^  +  y'  =  a;    (:r  +  y) (.'uH 2/')  =  ^ (^'^  +  2/0- 

86.  (a;' +  y')  (^  +  2/')  =  «^2/ ;   (^' +  2/)  C^'' +  y*)  =  ^^y- 

87.  Or  +  y)X.i'^  +  2/')  =  a  ;    (.r^  +  2/7  C^*  +  2/*)  =- ^. 

88.  (x  —  y)  (x"^  —  y^)  (x*  —  y*)  =  4  rt^ry  ; 

(.^  +  y)  (:.-^  +  y^)  (x'  +  f)  -  Z» {x  -  y). 

89.  a;^y  -\-  xy^  =  a  (x^y  -f-  xy^)  =  b(x*-{-  y*). 


/» =  a"  -  U'. 


)'(«*  +  ''*)• 


%xy. 


1  + 


)  =  hxy. 
b. 


RIMUT.TANEOTT.S   QI:A  DRATICS. 


90.    a  (r^  ^}-  /)  -  ab  (:r  -  h  y)   -  />^'y  (.f'  -  h  /)• 


91. 


?/      a-^ 


b\    x^  —  if_a!'~h^ 


x' 

x'^  —  7/"      a'  —  ^^ '    X*  —  i/*      a*  —  i* 


92.  ^"  +  .y\^«H^>\  ^'±j/_^f^'. 

a;"  —  ?/      a'  —  Z*"^ '    cc^  —  tf      a"  -  (6'' 

93.  af*  ~  2ax  ~  It/  ]   if      2 ay  —  bx. 

94.  (.t^  +  y)Ct"'  +  y')  =  «;  (.r-y)(.r-''-?/')^^>. 

96.    (:r  +  y)(r''  +  ?/')  =  aa;?/;  {:c  -  y){r' —  if)  ^  hxy. 

97.  (.t4-y)(^-''4-y'')  =  «C^'+?/) ;  (^•-2/)(^''-yO  =  *(^^+3/'0- 


98. 


(a;+y)«(r^+7/) 


(a.''^+^-y+y^)('^-N-2/"0 


=  «' 


(a;-?/)'(r'-?/') 


3 


=  //. 


99.  il±t)i':±lf  =  2a?-  (^-'-yX^-y) 


100. 


=  8a'^ 


{:^  -fKx-yf 
(jic'~xy-\-yy 


W 


Sb' 


101.  'xy(x  +  y)(r'-'rf)  =  c(,\   xy(x~y){x^-f)^b. 

102.  :r(:^:  +  y)(a7+27/)(a;  +  3y).-a^  (^^,jy^(^^^2yy  =:  b. 

103.  (:r+l)(y-l)  =  a(:r-l)(y  +  l); 
(:r^+l)(y-l)^  =  i^(:r-l)\y^  +  l). 

104.  x  +  y==a(l+xy);    (x -]- t/)' =  b\l -^  xY). 

105.  .T  +  ?/  =  a(l  +  :ry);    .r' +3/^ -- Z/'^l -|-.^y). 

106.  (x  +  y)(y-l)  =  a(x~l)(y  +  l); 
(^^-l)(y-l)=.b\f-l)(x~l). 


25G 


SIMULTANEOUS   QUADRATICS. 


109. 


110. 


112. 


113. 


114. 


115. 


116. 


117. 


118. 


119. 


120. 


107  (L-M)(i-i-y;,,^,,.  (1  Mr(i  i-.y;v_-& 


(1  -.r^)(l-y^) 


108    (^•f--^)(g+y)_,,.  (^•Mi^*)(^i±y*)^A 


(x  —  in)  (y.—  v)         '    (a;  -  ?/i)*(y  —  nf 
(.r+l)(?/+l)__«.  (x* +!)(?/+ l)_^ 


1  +  a;  k 


1  +  y 


g 


^(1  +  y  +  y'O 


.r(l-fy'0 


:r(l+7/-0 


=  a 


.rXl+7/«)        • 


(a:  —  y) (.■?:?/— 1)        2ab    '  y{x^—V)      a-\-b 
x^y)(l-\-xy)  ^  ^2 ^^2 

3:4-?/)(l+^.?/) 
.r-2/)(l-a.-y) 

a;  +  ?/)(l-f  n-y) 


a 


x-y){l--xy) 


a 


« 


(.r^  +  y^)(l  +  ^-y)_^ 

(.^••^-y•0(l-:^■y) 
(^  +  y)(l  +  a:y)  ^  j_ 


SIMULTANKors    lifA  DKATK'H. 


257 


121     (£±^Ml±Jll  ^  a  '   Cg'  +  ?/)(l-hrV),_5 
•    (.v^y)(l-xy)         '   {x^     y'){i--ry) 

122  (•"'  +  ^y + y-lKL+^i^-h^!^)  ^  ^  • 

(a:  +  ?/)Xl+.^y)'^ 

Cr'  -  ary  +  ?/')  (1  —  xy  +  .rV ) .    , 
(x-y)\l    -xyy 

123.  .r* -3a;'?/  +  5a'''^  +  ?/'    -0;    ?/ -    .7;V/      2 a'^o; --  0. 

124.  2a:(y'  -  2a:)'  =  a  ;  y(/  -  2.r)V(y'  "  '^^)      ^^• 

Hence,  deduce  the  solution  of  x^  —  5a;'  +  2  =  0. 

125.  2xy(x''-{-y'y=a]   (a;*' -  y')  (a;' +  y')' -=  6. 


Ex.  63. 


2.    x"^  —  yz 
?/'  —  xz 

2 


1. 

2. 


1.    (2.rfy     42)(:f  +  y  +  2)=:24, 
(a;  +  2y-2z)(a;  +  y-h2)  =  G, 
(-2a;4-  3?/  +  5z)(a;  +  y  +  z)  --  30.  2'  -  a:y  =  3. 

3.  (a.'  +  2y-32)Cr  +  y  +  2)-2(a:y  +  y2-h2ar)--12, 
(2a;  ~  3?/  -f-  2)  (.r  +  y  +  2)^-  (a;y  +  y2  +  zx)  =  61, 

(3  a;  —  y  +  2  2)  (a;  +  y  +  2)  —  5  (a;y  +  y2  +  2a;)  =  5. 

4.  a;'-y2  =  0,  5.    (x^ J\-y^^zJ +  (x+ 7/^  =  31, 

x  +  y  +  z=7,  (x'  +  y'i-zy  +  (x-]-y+zf=:12d, 

x'  +  7/  +  z'==2l.        (x  +  yy-h(x  +  y  +  zy  =  Sl. 


6.    x-\-yz      14, 
y  +  2a;=ll, 
2  -f  a:y  _  10. 

8. 

X  +y  _5z, 
^'  +  3/' --392, 
a;3  +  y'- 105  2^ 

7.    a;  +  y  — 82, 
a;'  +  y'-134 

x^  +  y"^  -\~  z^  — 

z\ 
134. 

9. 

a;  4-y  -72, 
ar'  +  y'  =  252', 
a;*  +  y*-6742^ 

258 


SIMULTANEOUS    QUADRATICS. 


10.  X  +y  =72;, 

•T^ -fy^-:  20,272  z. 

1 1 .  x-\-y :  y  ]-z :  z-\-x  •.•.a:b\c, 
{a-\-h-\-c)xyz  =  'l. 

1 2 .  x-\-y :  7/-\-z :  2  f  :c  wa-.h-.c, 
{a-\-b^c)xyz  -=  2  (.^•+?/-[-2;). 


14.  zi'i^y. 

\Z         Xj 


=  a, 


xni-\- 


y> 


15.     (y  +  2)(2rr  +  y  +  2)  =  «. 

(2  +  .r)(:r  +  2y  +  2)-/;, 
(a;+y)(a;  +  ?/  +  22)-=c. 


13.  (x'\-y  —  z)x^=  a, 
{x-yi-z)y  =  b, 
{—x-\-y-\-z)z^c. 

16.  x{y  -\-  z)  :  y  {z  -Y  x)  \  z{x  -^  y^  -^h  -\-  c  :  c  -\-  a  \  a  -{-h, 
xy  -\-  yz  -\-  zx  =  {a\-h  ^  c)  (a;  +  ?/  +  z) . 

17.  {a-^h)x  +  {b^c)y^{c-\-a)z  =  {a'\-h^c)Q€^y-\-z), 
a{X'\-y)  =  c{y-^z), 

{x  -+•  y)^  +  (y  +  2)^  +  (^  +  ^^0'  =  4  (a^  +  ^»^  +  c'). 

18.  c(:^:  +  ?/)  +  ^'C2;-2)-a(y  +  2)  =  0, 
^,(0;  — z)  =  (a  — c?)?/, 

^'  +  2/'  +  2'==«'-f^'  +  ^'. 

19.  x-\-y  —  az~'X~by-\-z  —  ~cx-\-y-\-z-—  xyz. 

20.  (a  +  ^>  +  c)  {x-y)-^a{x^z)  ~b{y-\-  z)  =  0, 
(a  +  b  +  c)  (x  —  2)  +  a(x-^y)  -  c{y  +  2)  =  0, 


ax^ 


+ 


i^ 


+ 


6?2' 


{b^-cy      {c-\-af      (a  +  ^) 


1, 


a; 


y 


c. 


21.  .•?:?/  +  -  =  «,     ?/2-!--  =  o,     2a; +- 

^      z        '     -^     '  a;        '  '  y 

22.  y -\-  z  \  z  -\-  X  :  X -{- y  \  \  b  -\-  c  \  c  -\-  a  \  a  ^b, 

\x  +  .y  +  2)  (^y^;)  =  (a  +  Z>  +  c)  (.ry  +  2/2;  +  zx). 

23.  x^  —  yz^^a,     if  —  xz  =  b,     z^  —  xy  ~c. 

24.  a;'^+(y-2y  =  <     ^^+(2-.^^-^'^     .^^(^^-y)^^^^^ 


SIMULTANEOUS    QUADRATICS. 


2/  +  2)  =  «, 
\y-{-Z)  =  h, 


xyz. 

0, 
0, 


2/y 


25 .  x^-}-  xy  +  y'  =  a^   y"^  -\- yz -{- z^  --  h'\    z^  -f  ^a;  +  ^^  —  <^'' 

26.  a;  +  y*  —  z'''  -|-  3 xyz  ==  a(x  -{-  y  —  z), 
^  ~  y'  +  2^  +  3 a:yz  =  b(x  —  y  -}-  z), 


27.    a;  +  ?/  +  2a2;  =  0, 
^"  +  y**  +  2"  =  c". 


34.    (:r  ~  yf  =  az(x -{-y), 
x'  —  y'---=bz(x-{~y)\ 


28. 

:r  +  y  —  az  —  0, 
y(x'  +  f)      h\ 

35. 

x  —  y~a, 
u  —  v  —  b, 
xy  —  uv, 

29. 

x{y-l){z-\)-- 

=  2a, 

x^—y^-\-u'—v' 

x\f-l){z'-l)-- 
:r^(/--l)(2^-l)  = 

=  Uz, 

36. 

X  +y  —  a, 

u  -{-  V  —  6, 

30. 

x{y      l)-a{z 

-1), 

X^  -\-y?  -—  c^, 

x\y'-l)-b'(z'- 

-1), 

y'^v"-  e\ 

a^yf     l)-c\z'- 

-1). 

c(a-\-b). 


-,., „7 


31.  a;  (y  —  1)  =  a(z- 1), 

x\y'-l)=-c\z'-\). 

32.  a:  (y  —  1)  =  a(2  — 1), 


37.    xy=uv  =  a\ 

X  ~\-y  -\-u-\-  x^=b^ 
or*  -f  if  -\-y?-\-v^  =  0^. 


38.    :i'y  =  uv 


a' 


x 


X' 


x-\  y^u-\-v  =  b, 


X 


4  ^4 


-^-y  -\-u  -{-V  =  c 


33.  a;  (?/  —  1)  =  a(z  —  1),         39.    xy  ~  uv 


a' 


x\f-\)  =  b\z^~\\ 


a- 


-^y  +  u  +  v  =  b, 


x'  -{- y"  -{•  w  -{- v 


5    I    «.5  .5 


200 


SIMULTANEOUS    QUADRATICS. 


40.    xy  -  uv  ~  a^, 

(x  +  uf  4-  (y  +  vf  -  c\ 


41. 


xi/  =  uv  =  a^, 

(x-^uy  rOj-\-vy  =  c\ 


42. 


xy  =  uv, 

X  i-  y  +  u  i-v 

X^  -|-  y2  _|_  ^2  _j_  ^2 


43.  xy  =  uv, 

X  -\-  y  +  ?/ -f"  "^ 
x'-i-if  +  u'  +  v^ 
x*-^y*-{-u'-\-v' 

44.  xy  =  uv, 

X  -\-  y  -]-  u  H-  V 
x^  -f  y^  +  ^^^  +  ^^ 
x^  +  y^  +  ^^^  +  ^^ 


a, 


c\ 


—  ^4 


a, 


45.    xy  ^=  uv, 

^  +y  -{-u  -\-v  ■=  a, 
x^-\-y^-\-u''  +  v^  =  h\ 


sc'-{-y'-[-u 


*  -i-  V*  =  c\ 


46.    :r?/  —  i/v  =  0, 
xu-\-yv  =  a^, 
X  +  ?/  +  zi  +  ?;  =  ^, 
^.3  -\-y^-{-  u^  -{-v^  =  c^. 


47. 

x'  -f-  y2  ^  ^2^ 

2^2     _|_     ^2  _  ^^2^ 

wa;  +  -yy      e^, 

vx  -\-uy  -  7i^ 

48. 

•'^'  f  -  y  -[-  u-\-v 

xy  -\-uv      l)\ 

a. 


x"  +  y2 
?^^  +  v"^ 

49.    y(l  +  .T^)-- 
m(1+/) 
Kl  +  w^) 


7>i' 


7ll 


2:r, 

=  2w, 

=  2i;. 


50.    X  ^y  -\-  u-\-v~  a, 

{x  +  yy-^{u-\-vf=^h\ 


51. 


.r 

_2rt 

—  u 

y 

-  -  z 

■2u 

2 

V 

2^> 

—  u 
■2u 

2 

_2c 

~u 

x-\~y     c  —  2u 
x'-\-y'-^z^-=e\ 


SIMULTANEOrS   QI'A  DRATICS. 


L>H1 


52. 


53. 


54. 


55. 


— • ' —  ^=  a, 

1  +  y  +  f 

li-x  +  if 


X 


+  1 


y  +  1 


a 


-1 


(1    -.r)(l-y)      1     -a 

(l  +  :i-)(l-y)__l  +  /> 
(l-.r)(l  +  y)      1    -/. 


X  -y  y  __a^  —  g'^ 
1  -f-  .ry      a'*  +  ^ 

x-y   _  Z>'^  -  )8'^ 


1  -  .ry      6^  +  P' 


56. 


^  +  .V 


57. 


58. 


59. 


1  + 

:ry 

X  — 

-y 

1- 

xy 

x~- 

■V 

1 

xy 

.r — 

■y 

1  + 

xy 

1  + 

xy 

xAry 

1- 

xy 

X 


+ 


+ 


a 

h-Vc 

h      c 

a 

Ida 

a^  -  -  a^' 

2/;^     . 

b''      P' 

X  +  ?/  _ 

_2a 

1  +  xy 

711 

^^-y  _ 

2h 

.  c 

~  y     1  -  ^y 


71 


a. 


^'(i  +  y"0 

xil-f)        ' 


60.    2aa;-=(/^-f  <?--«)(?/-[- z), 
2Z*y-(c  +  a  -/>)(x  +  z), 

(^+y+2)'4-^'*Hy'+2'-4(a'^f&HO- 


61, 


62. 


a: 


a 


y 


1      &-1 


a-^ 


y 


-I      b' 


x^j-xy+y"^  ^  :r'+y^  ^  £y. 


a;  — ^' 


'y+y' 


a 


64.    r'^  +  y'^ 


a;'^y  —  xy"^ 


X 


a 


x~y 


x-\-y 


65.    xy-Y-^aix"  ~  y'), 
xy-^-~b{x'^if). 


63.    .9;*  +  a;y  +  y*  =  a,  66.    r' --=  a(^'^+y'^)  -%, 


.'T'  -f-  xy  +  y'  -rr  Z>. 


y'''^-b{x'-^y')-axy. 


262 


SIMULTANEUUS    QUADnATKJS. 


•v 


•m 


67.    4<:r'  +  l)  =  (a  +  />)(.^•-2/)^ 


68.    x'-n'  ^-^'^'^^"' 


T^(^^'  +  -^'!/  +  f)(^  +  y), 


69. 


X  -\-  x'^ 

X  +  y' 


a, 

b. 


70.  ^^X^^«, 

71.  x{y  ^  z)  —  a^ 
y{z-]-x)---b, 
z{x-\-y)--=c. 

72.  Oi-  hy)(^  +  2)  = 


<2, 


73.    a;(a;  +  ?/-f  z) 


a 
h 


6'  —  xy. 


74.    .r'  — (y 

-  ^)^      «, 

2/-^-(.- 

-xy.-..h, 

2^       (a;- 

-  yy  -  ^• 

75.    .6-'^  +  /- 

-az, 

.r  +?y  = 

-hz, 

X 


V  —  cz. 


76. 


11 

x'       f 

2a 

1     1 

2h 

x"-       f 

--&' 

1     1 

_  1 
c 

77.    a;y 


.+  1 
f){z~VVj^-=Uz. 


(x' 


78 .    Find  the  real  roots  of  the  system  of  equations, 

x"^  -f-  ^v"^  +  v''^  =  a^  vw  +  "f^  (y  +  2)  ~  ^^, 

ty^  +  y'^  +  w^  =  &^  tvu  -\-  V  (z-\-x)  =  ca, 

v^  -f-  V?  +  2^  -^^  (?^  uv  +  7(;(.i--|-y)  —  a6. 


...  -v-  '^: 


hy)- 


r  =  a, 
)'  -~-  h, 


ia 

?' 


-1)= 
+17 

2  a, 

ions, 

-2/)  = 

^-  he, 
=  ca, 
-  ah. 

CHAPTER  VIII. 


Indices  and  Surds, 


§  50.    The  general  Index-laws  are  : 


an  X  an  -=  a«'  ?, 

m  ;)  in     p 

a«  -^  ai  ~-  a»    ?, 

W  TO  W 

(ab)n     =  a«  X  />», 

m  m  VI 

(a-~h)n:==  a'n  -f-  bn, 

m  p  mp 

(a 't)?      =^  a'^1 . 


(1) 

(2) 
(3) 
(4) 
(6) 


The  law  connecting  the  Index  and  Surd  symbols  is 

The  indices  g,  ;J,  J,  etc.,  are  generally  used  to  denote 
"  either  square-root,"  "  any  of  the  cube-roots,"  "  an}'"  one  -  f 
the  fourth-roots,"  etc. 

The  surd  symbols  -y/,  -y/,  -y/,  etc.,  are  by  some  writers 
restricted  to  indicate  the  arithmetical  or  absolute  roots, 
sometimes  called  the  positivf^  roots.     Thus, 

V4  =  2,  but  42  =  ±  2. 

.•.4^  =  ±V4. 

V[(- 2)'^]  =  V4 -^  2.  ,     ^       .     ^, 

Sy27  =  3,  but  27i  =  3  01  3  f~^  ;''VJY 

.•.8^  =  (l^)sy27. 

^16  --  2,  but  16^  ---  rb  2  or  ±  2i. 

.•.16i-(ll)</16. 


Also, 


264 


INDICES    AND    STRDS. 


With    this  restriction,  the    general    connecting    formula 
would  be 

an=:.(ln)y(a"'). 

In  the  follov/ing  exercises  this  restriction   need   not  Ijo 
observed. 


Ex.  64. 

1.    What  is  the  arithmetical  value  of  each  of  the  following: 


36^;    27*;    16^;    32^;    4'^;    8^;    27^;    64^    32^; 
64^  ;    Sr  ;   (3|)*  ;    (5,^)^  ;    (lA)^  ;    (0-25)^  ; 
(0.027f  ;   49"-^    32«-^ ;    81"-^^? 


-h 


2.  Interpret  6r=^ ;  a°;   c/ ;    (a^)'';  ci^  ' ',  «     ;  (a  ^)  ^ ; 

a  \  a    . 

3.  What  is  the  arithmetical  value  of  36"^  ;  27~* ;  (0.16)"^ 

(0.0016)-^  ;   (i)-^  ;   (^)-^  ;    {^^  ;    {bi^fh 

4.  Prove  (a"*)" -=  (a")'^  ;    (a'")"  =  (a")'"  I    a"*"  =  (a" ')'^  ; 

and  express  those  theorems  in  words. 

G,    Simplify  a*  X  a*  ;    c^  X  c^  \   m^Xin      ',    n*  X  n"^^  ; 
(71)*  X  (2f )^  X  mf  ; 
*7^       r^        r/*        p"^"^        }'  a  3  11 

V  T  i'  V'  ^i'  (2rx(oi)'.-ar'*. 


rr 


5 


.'I' 


6.    Remove  the  brackets  from 


:?nU.  /-ki 


.--!l 


{ay-;    ih)';    («')=;    (rf^)'' ;    (.-')';    (/'^p ; 


-I  3.     6    XO  _1  2         1  •?         4        n 


ng    formula 


aced  not  ha 


be  following: 


P/iS 


64=^;   32^; 
).25)^  ; 


(«  )   ; 


'-*;(0.16) 

16/' 


-I 


In^  X  n  ^^ 


11)^'  -  ii)-''- 


INDICES    AN!)    SURDS. 


205 


7.    Remove  the  hraekets  and  simplify 


X 


X 


X 


X 


8.  Simplify    -x[x~-(-x)  'f;    ■r[(-x)~^{- x)-']~^', 

{-x)-\x-^x~^)\ 

9.  Determine  the  commensurable  and  the  surd  factors  of 

12^'2#;    18"^;    (-81)^    12^;    04^;    (Jj^ 

(6f)~l 
The  surd  factor  must  be  the  incommensurable  rooi  of 
an  integer. 

10.    Simplify   d^  +  18^-50^    72^  +  (j'-^f  -  (ihT^  > 


>i 


.i 


h         .        o^ 


>^         oh 


[(6  +  2^)  (6  -  2')Y  ;   (2^  +  3^/  +  (2^  -  3^) 


1     1 


2n5 


(2^  +  3^)  (4^  +  9'  -  6')  ;  (7'^  -  3-)-(7-  +  3^) 


-'  }  2 


S[(a  +  .r)(:r  +  ^)r-[(^^-^)('^'-WI 


inJ 


H 


Express  as  surds : 


11.    a    ]    X   ]   p 


h 


TO-3 


12.  a;"+i;    ?/-**+3;    a"'^    />"+«. 

a  TO- 

13.  (aa7-Z»)»^;    (x^  -  i x -\- 1)^^  ;    (p-qxY'l 

Express  with  indices : 

14.  ^a':    -^c';    V-t'''^    </'!/'"''',    V(^''0 ;    V^*"'- 


2G0 


INDICES    AND    SURDS. 


15.  y^c'^iln;    ^{(e-l-li'f-   [-^[jt^\-i;')f-   .^\(„.     h)x]: 

16.  {cH^f  ;    {h~'^)-^  ;   (."^)"*  ;  {^Y^  ;   (air)"*  ; 

Simplify  the  following,  tixpressing  tlio  results  by  both 
notations  : 

17.  a  X  a  '^  ',  a^X  a  ^ ;  cr  X  (('^  ;  a  X  a  ^  ;  a  ^  X  -^a  ; 


3 


3 


18. 


a 


i 


.?r 


a 


^6'\    -,y^-\    ^y~\    sy(24«-^). 


Vc^'  sya'  -sy^ '  M  '    -*      27a 


y 


e(ahy  — 


ac 


1 
be  —  c  (ah) 


19. 


1  -1  3.  _3  3n  3n 

rt"  -f-  a      .    a^  —  a  ^  .    a'li  —a-i^    a^  -\-\-\-  ar'^ 


a  —  a         a  —  a         a   2  -f  a  2  '       ' 


20.    Divide  x ~yhj  Xn  —  yn  ; 

:c'  +  oTx^  +  a^  by  .7;   -f-  aV  +  a^  ; 

^  +  y  +  2:  -  ^a^^^  by  rr*  +  y*  +  2*  I 

2a^>  -\-2hc  +  26'a  -  a'^  --  h'  -  c"  by  a^  +  i*  +  c^. 


I 


Ex.  65. 

1.    Express  the  following  quantities  (i.)  as  quadratic  surds  ; 
(ii.)  as  cubic  surds  ;  (iii.)  as  quartic  surds  : 

a\  3a;  2d^  \  d^x\  x^]  y^  \  a"'";    -;  mx  p;  0.1; 
0.01;   \lx\ 


m . 


[(,(,■  -h)x]; 


Us  Ly  l)oth 


X  V^' '' 


1  . 


-2 


a 

+  ,7' 


6*  +  c\ 


idratic  surds ; 
|?a;~i;  0.1; 


INDICES    AND    Sl'RDS. 


267 


2.   Reduce  to  entin^  surds  : 


yV.  ^' 


^{xhr') ; 


^MC^)^  (-^+^(^7:9^  :-^!>((i^j^ 

{x  - yy^  ^(:r  -f-  2.ry  +  /j"^ ;  (.f  -  x'')^{x'^  +  1)^ 
3.    Reduce  to  their  siraplest  form  : 

V12;  V8;  V^O;  -v/iC;  4s>'0.250;  vi;  -x/i;  VA; 
5sy(-320);  -</(!- hV);  V'^^  V("'^-'0  ;  -v/«^ 

3isy(54.r«);  -^(.rVV);  VK(1-^')];   S^L^H^^' -  1)1 ; 

V(«^);  V"""^';  -v/«"+";  V«'""^';  V^^""~';  V(^^'^'  +  «'); 

^(a-'  +  2  «^i'  +  «V)  ;   ^[{x  -  1)  (.r^  -  1)] ; 
^[{d'  +  2  a.r  +  o;^)  (a"^  +  x')] ;   -^[(^^-^  -  dj{x  ~  a)] ; 
V'(4.i''-8:r'  +  4:r);   V(8^' -  16a;  +  8)  ; 
Si/[(:r'^  -  2  +  a;-2)(:r*  -  2  .^^  +  1)]  ; 


^ 


2a;-2+2a:' 
x-\-'2-\-x 


-1 


xld 


.r" 


■6a:'+3.^ 


7:c^+18a;+3 


)^a1 


(a'-a^>)^+4aV; 


Cfc 


-^» 


4.    Compare  the  following  quantities  by  reducing  them 
to  the  same  surd  index  : 

2:  V3,  2:sy9;   V^  :  a/3  ;   ^aO-.-^m-,  2V2  : -^22  ; 


2G8 


INDKJES    ANIJ    SURDS. 


6.  Reduce  to  simple  surds  with  lowest  integral  surd 
index : 

</«/^'')\  s:/(V27);  V(sy8i);  </(^81);  v(^V«) ) 
s/(f.V^O;  V('^a/-0;  s^C^'^^-^O;  sl/C^V''^);  V(3sy3); 

6.  In  the  following  quantities,  combine  the  terms  involv- 
ing the  same  radical : 

3V2  +  5V2-7V2;  V8-V2;  syiG-h3^2; 
S^IG  +  V^ ;  rt  V^'  ~  V-^' ;  ^\/^'  —  ^  \/^ ; 

8V«  +  5 V^  -  'J' V«  +  V(4«)  -  3 V(4^^■)  +  4 V(9:^)  ; 
V^  +  3  V(2:i')  -  2V(3^)  +  V(4^)  -  V(8^^')  +  V(12^) ; 
7 :c  -  3  V^  +  5-^a-- -  2^.x'' +  ^:^^ ; 
4  V(^fc'^)  +  2^(b'x)  -  3  V[(«  +  byx] ; 

V[(«  -  ^)':^^] + V[(« + ^)''^']  -  VCf^'-'^) + V[(i  -  «)'^']  -  V-^ ; 

^(a  ~h)+  V(16  a  -  16  Z^)  +  V(«^'  -  ^^')  -  V[9  («  "  ^)] ) 

V(^i'  +  2a^Z>  +  ai^)  ~  V(«'  -  2rt'^>  +  ah')  -  ^(^iah'). 

7.  In  the  following  qiia,ntities,  perform,  as  far  as  possible, 
the  indicated  multiplications  and  divisions,  expressing  the 
results  in  their  simplest  forms  : 

V2XV6;  V3XV12;  Vl4xV35xviO;  V^ixV(3^0; 
V6^xv(l2^);  -V(^x)x^(8x);  V/xVy';  -v/y'x^/; 


INDICES    AND   SIRDS. 


20'.) 


tegral  surd 

^(3sy3) ; 
A-) ; 

;erms  involv- 

2; 


v/[9(«-^)]"' 

■rir  as  possible, 
pressing  the 


.A  -f-  v^'^"  ' ;  ('''  --  V^*"  "* ;  («  +  '0  -^  V(^*  ^-  ^') ; 
(a' -  ^-'O  -^- v(«  - •'^'0 ;  (^' - 1)  ■■- -y/i'^'  + 1)' ; 

(3 V«     5 V2  +  Vl^"^  -f-  V32  +  V '2  -  2 V^O)  X  V2 ; 
(7  V2  -  5VG  "  3V8  4-  4  V20)(  Vl8) ; 

(V^  +  VSjCV^-V''^);  (V2+i)(VG"V3); 

(3  -  V2)  (2  +  3  V2) ;    (5  V3  +  V^)  (^  V2  -  2) ; 

[V(-r+  i)+V(^-  i)][V(^  + 1)  -  V(^-- 1)] ; 

[ V(3«  - ^')  -i  V(3^  ~ «)][ V(3«  - ^) -  V(3^ - «)] ; 

V(« + V^)  X  V(«  -  V^) ;  V(^''^' + Vy)  X  V(  V'^*  -  V^) ; 

V[«  +  V(«'  -  ^')]  X  V[«  -  V(«'  -  ^'0] ; 

^[x  -  VG^•^  - 1)]  X  ^[.r  +  V(^''  - 1)] ; 

V(8  +  3v7)xV(8-3V7);  (V«  +  V^)';  (V«  +  V^)'; 

[a  +  v(i  -  «')]'  •'  [  V(«  +  ^  -  ^'^0  -  V('^  -  ^  +  ^)7 ; 

\[-V(a-\-x)ix-h)]{-^[{a~x)(x-\-h)]\-^; 

S  V[(«  +  ^^0  0^  +  i)]  +  V[(^^  -  ^')  (^^  -  ^)]  r ; 


"^(fef)-All 


X 


a  —  a;, 


v(*-'  - 1) 


\ 


'^i 


.r 


) 


(^«+^i)» ;  [V(V«  +  Vi) + V( V"  -  V^')]' 


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^^ 


^\ 


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& 


■  I:- 


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m^ 


270 


INDICES    AND    SURDS. 


[V(Vlo-M)-V(Vio -Df; 
(V«+V^+v^)(V^+V^-V«)(V^+V«-V^) 

(^l+^|!+')(^l^^lI-■) 

( ^a  +  -^i  +  S^c)  [  ^a^  +  </b'  -  ^c"  -  2^ {be) 

-2-^{ea)-2^{ah)].  . 

8.  Find  rationalizing  multipliers  for  the  following  expres- 
sions, and  also  the  products  of  multiplication  by  these : 

a  +  V^''    V^  +  ^V^;   a-yjh  —  h^a-,   a-\-^{a^  —  x^)] 

V(a  -x)-  v(«  +  ^) ;  V(«'  +  V^)  +  V(«'  -  V^) ; 

V[8+V(24+V5)]-V[8+V(24-V5)]:  V«+V*+V^; 

34-V2+V7;  V6+V5-V3-V2;  v«+VHVH-V<^; 

v(i+a)-va-«)+va+^)-va-^);  >+-v/^; 

^a'-^c';  </a+>/^;    N/a--v/^;  </«+V«;  V^+^^y'; 

9.  Rationalize  the  divisors  and  the  denominators  in  the 
following,  and  reduce  the  results  to  their  simplest  form : 

1^(2-V3);  3-^(3  +  V6);  5-^(V2  +  V7); 

(V3  +  V2)-^(V3-V2);  (7V5  +  5V7)-(V5  +  V7); 


INDICES    AND    SURDS. 


271 


-ah^'2,^-h') 


-V^) 


4 

[lowing  expres- 
1  by  these : 

a;   V^+^2/'; 

linators  in  the 
Lplest  form : 


a  -^  {-y/a  +  a)  ;    (^  —  «)  -^  ( V-^  ~  V«)  J 

(a^  +  a6  +  b')  ^  [a  +  V(«^)  +  ^]  i    (x  +  o)~-{ ^x  +  -</a) ; 

riV^  +  ^>V.y.  2V6  1  +  3V2-2V3. 

V6-V5-V3  +  V2. 2 . 

V64-v^'^~V3-V2'  V(«+l)-V(«-l)' 

2£ .    ft  f  ^  +  VCa'  +  a:')  . 

-^(a  +  c)  +  V(«  —  <?)  '    «  +  -^^  —  VC*^"^  +  ^)  ' 

V(a  +  ^)  +  \/(ft  —  X)  1 


V(i  -  ^0  +  V(i  -  «0  '  a V(i  -  ^") -1- ^ V(i  -  "') ' 


a4 


V 


(1 +  «)(!  +  &) 


(l  +  a)(l  +  i) 


ZlV 


+v 


(l-a)(l-^>) 


(l-a)(l-6)] 


(g  -  .t:)  V(^^  +  ?/)  --  {h  -  y)  V(ft'  +  x^)  ■ 
(g  +  :^)  V(^'  +  f)  +  (^  +  3/)  V(a'  +  ^'0  ' 

yq  +  g)  -  V(l  -g)+ V(l  +  ^>)- V(l  -^>) 
V(l  +  a)  +  V(l  -  «)  +  V(l  +  ^)  +  V(l  -  -  *) 

V(a:  +  g)  -  V(^  -  ft)  -  V(3:  +  ^>)  +  V(a^  -  ^) 

V(^  +  «)  +  V(^-«)  +  V(^  +  ^)  +  V(^-^) 


g 


1     1 


g  +  V^ 
-y/x 


+  V(ft'-1) 


V(«^-l) 


V^    V 


a? 


■sjx      V3/      V-'g      V' 


g 


+  -77. 


V<^  I  V 


+ 


07 


V^    Vy    V^    V' 


a 


* 


;* 


272 


INDICES   AND   SURDS. 


10.    Find  the  values  of  the   following  expressions  for 
71  =  1,  2,  3,  4,  5,  respectively  : 

1    r(2  +  v'6r+'-(2+V6)  _  (2-V6)»-^^-(2-VG)-| 

2V6L  l  +  V^  l-^Q 


'] 


11.   Show  that 
1 


i[a:4-v(^-i)r-'+[^-v(^'-i)r-*^2i 


2(^-1) 

is  a  square  for  w  =  1,  2,  or  3,  respectively. 

12.  Extract  the  square  roots  of : 

a;  +  3/  — 2V(a^);   a -\- c -\- e  +  2-y/{ac -\- ce) ] 
a  +  2c  +  e  +  2-y/[{a  +  c)(e-^e)];   2a  +  2 V(«' -  c') ; 
2[a^  4-  b'-^(a'  +  a'b'  +  ^>*)]  ;   x-2  +  x-' ; 
V^+2  +  V^"*;   x  +  Sa^-\-3r'-{-2x^x-i-2x''^x; 

2a;+V(3a;'-f-2:ry-y');  5-2V6;  10  +  2V21;  9  +  4V5; 
12~5V6;  70  +  3V451;  4- Vl5;  4-V15;  7+4V3; 
9  +  2V6  +  4(V  +  2V3);  15.25- 5  V0.6. 

13.  Find  the  value  of: 


(a-^b)xi/ 
ay^  +  hx^ ' 


given  X 


_     qV( 


a 


V(«  +  V) 


and 


y 


1:\/1 


V(«+-) 


V(^'  +  2/').  given  a;  ---  ^{ac),  y  =  ^(a'e) 
^^  +  ^(x'  +  1) 


given  X 


X  -  V(^''  +  1) 
^(li-^)-^(l-x) 

va+^)+v(i-^)' 


-IW^  \«) 


given  a:  = 


a' 


2ab 

+  b' 


ressions  for 


(2-VG) 
6 


5 


COMPLEX   (QUANTITIES. 


273 


given  x^= 


2(^l^'^ll 


2aV(l  +  a:') 

14.  li^{x-\-a-^h)^y/{x-\-c-\-d) 

15.  Simplify      Ml+V5)-^'-2  i(l-V5).--2 


-j4«±i^2! 


-  O ; 


V^' 


;  9  +  4V5; 
|5;  7+4V3; 


^\' 


V) 


Complex  Quantities. 

Quantities  of  the  form  a  +  &^— 1,  in  which  neither  a 
nor  h  involves  -^—V,  are  called  Complex  Quantities.  The 
letter  i  (orj)  is  frequently  used  as  the  symbol  of  the  diten- 
sive  unit  -y/—!,  so  that  a-f-i^—l  would  be  written  a-\-bi. 
So  also  -yZ—x  =  i-y/x,  -yj—x  X  y/~  ij  =  i'^^{xy)  =  —  ^xy, 
and  i'  =  — I. 

Ex.  QQ, 

Simplify  the  following,  writing  i  for  y'—  1  in  any  result 
in  which  the  latter  occurs : 

1.    V-4;   V-36;   V-81;   V-8;   V-12;  ^-12- 

■</-8;  V-5x  V-6;  V-6XV-8;  V---8XV12; 

V-8  X  ^-8 ;    V-  5  X  V-20. 

2.  V-^;  V-^';  V-a';  V-«''*;  V(-«)'; 
V-axv— 1;  V^xv-«- 

3.  ^■^    i';    z*-    i^    i»;    z'^    i'^;    i" ;    i>«;    i*";    i*«+» ; 

4.  at  X  bi ;    z^a;  X  2-^2/ ;    ^  * ;    *  V^  5    z V"~«  i   **  V~" "'  J 


274 


COMPLEX    CiUANTlTIES. 


5.  v-»"; 

V-i'\  V-*'*;  V-^';  V-^■■'^  V-^"*- 

6     V~6. 

v'    6 .     v^  .     V<^  .  V    « .      1 

V3    • 

V-3'  v-3'  v-^'  v-^'  V-1' 

a             a' 

.  V(—  «^) .  -  V—  1 .      «'     . 

V-a'    V- 

«''     V  -^  '    V  «  '  a/   «'' 

V-^'"  v(- 

7     1.     1  . 

1-111         -1         1 

^3'         ^-       '       ^-6'    ^^n+l'      ^4H+1'      ^Mn-l' 

aH           X 

.       -y     .           oi' 

8.  v(«-^)x  V(^-«);  V(3^-42/)x  v(4y-3^); 

(3  4_5,)(7  +  4^);   (8_9^)(8-7^); 

(7  -  i-^b)  (7  +  i VIO) ;  ( V3  -  ^ V6)  ( V2  -  V^) ;    ' 
{a-\-hi){c-\-ci)\   [a-\-{a—\)i'][a-\-{a-\-V)i\] 

(y/a  +  e'V^)  ( V^  ~  * V^)  I    (^  +  ^0  (<^  ~  ^0  J 
(az  +  &)  (ai  —  b)  ;    ( Va  +  i^h)  {-y/a  —  iy/b)  ; 

{cty/b  4-  a'V^)  («V*  —  ci-s/x)  ;    V(l  +  0  X  V(i  -  0 ; 

V(3  +  40x  V(3-40;    V(12  +  50x  V(12-5z); 

(l  +  ^)^   (V«-^V^)^;   (5-2^6)^ 

(a  +  ^0'  +  (a  -  *0' ;    («  +  ^'0'  -  (a  -  ^0' ; 

(a  +  Uy  +  (ai  -  by  ;   [  V(4  +  3  0  +  V(4  -  3  i)] ' ; 

[V(3-40-V(3  +  4i)]^   [V(l  +  0  +  V(l-^)]^ 

{I -{-if)   (1  +  0*;   («4-^0*;   (a  +  5iy  +  (a-60»; 

(a  +  bif  -  (a  -  6z7  ; 


( 


i+iV3' 


f 


1  +  ysy.  /- 1  -  V3 


)■ 


k 


in 


1 


i-l  ' 


^(47/-3a;); 

jV6);  ■ 
]; 


/(I  -  i) ; 

2-5i); 


COMPLEX    (QUANTITIES. 


07  r 
^  t  o 


I'  C^)' 


{x  4-  zy)*  +  (a:  —  zy)* ;    (x  -f-  zy)*  -  (ix  |-  y)* ; 

( 1 4-  ^■  V5)*  +  ( 1  -  i V^)' ;  («  +  ^0'  ■  F  (a  -  bif ; 
(«  +  ^0=^ "  («  -  ^0=^ ;  ( 1  +  ^/  +  (1  -  O'^ ; 

(I -^  i^2f  +  {I  -  i^2y  ; 


B 


^Y 


ii[V(30-6V5)-l-y5]+ii[V15+V3+V(10-2V5)]!' 
for  all  positive  integral  values  of  7i. 


9. 


64 


21 


1  +  i^yS      1-1^7     A  +  Si-y/6     V2  +  ^V3 


IzL^Oi^.    1  -  ?' V3  .    1  +  ^    1  +  - 


1 


7-2zV5      l  +  iVS      l-'i      1  +  i      1-i      (1  +  0' 


1  4-^'' .    x-\-yi  _    «  +  ^V^' .    ?' V« 4-  V~ ^ 


a 


^>j 


1- 


X  —  y% 


a 


—  2\/:r      "y/—  a  —  z  ^h      ai-\-  b 


a 


+  eV(l  -  x')  .    V(:r-y)-V(y-a;).    _1 


n 


+ 


-zV(i-^"0    V(y-^')+V(^-y)    i  +  i    1-* 


i+i.i~i 


+ 


1 


l-i  '   l  +  i'    (l  +  i)'      (1-0       (1+0*      (1-0 
-\-yi  ,  a:  — yz .  a:+yi  _  x-yi  .   ^x-{-i^y  _  :^/]l±i^ 


4- hi 


hi    a  +  6i 


hi     -yjx  —  i^y      Y'y — *  V-^ 


J{\  +  «)  +  zV(l  -  g)      V(l  -  a)  \  /V(l  +  a) 


V(l  +  a)  -  iV(l  -  «)      V(l  -  «)  -  *V(1  +  «) 

10.  y(3  +  40  +  V(3-40;  V(3  +  40-V(3-40 

V(4  +  3  0  ±  V(4  -30;    V(l  +  2  V6)  ±  V(l  -  2  e  V^) 
V(5  +  2iV6)  ±  V(5  -  2iV6)  ; 


276 


COMI'LKX    ilV A NTITIKS. 


V(  V- +  V105)  ±  V(  V3  -  *  V105)  ; 

11.    Prove  that  both  i(     1  +  a V^)  ami  J  (- 1    -  i\/3) 
satisfy  the  equation 


or 


-0 


that  (a:  +  toy  -f  w'z)'  =  a;'  +  t/*  +  2'^ + 3  (a; + o>y)  (y + wz)  (z  +  coo:) 

and  that  (x-\-i/-]-z){x-i-<a^-{-io'^z)(x-]-io''2/-\-ioz)=ar^-\-y^-\-z^-~Sx^/z, 

in  which  co  represents   either   of  the   preceding   complex 
quantities. 

Hence,  prove  that : 

(i.)  [2a-b-c  +  (b~c) i^Sf 

=  [2b-c-ai-(c-a)  i  v/3J'-  [2c-a-h-\-  (a-h)  iy/2>Y ; 

(ii.)  i(?-\-i^ -\-iv^ —  Zuvw 

=  (a'  +  6'  +  c'  -  3a^»c)  (ar*  +  /  +  z'^  -  3a;3/z), 

if  2*  =  aa;  +  %  -[-  cz,  v  =^  ay  -\- hz -\-  ex,  lo  =-- a?  -{- hx -\-  cy, 
or  if  u=^ ax  -\-  cy  -Y  hz,  v=^cx-\-hy-^ az,  lu  —  bx  -\- ay  -\-  cz. 

12.    Prove  that  \[y/b  +  1  +  iy/{lO  -  2^5)]  satisfies  the 

equation  — -^==0. 
x-^\ 

Writing  w  for  the  preceding  complex  quantity,  prove  that 
(7  +  o)  +  0,'  +  3<o'')(7  -  CO*  -  w'-  3o>')  -  71,  and 

(^ + 2/  +  z)  (^  +  t'>V  ~  ^^  (^  —  <"'V  ""  *^^)  (a; — w2 + ui*z) 
X(x+ <x>*y + w'^z)  =  a;*  -I-  y*  +  2*^  -  5  ar*yz  -|-  6  xyV. 
Prove  that  [4«-}-(6-c)(V5-l)-f  (i  +  c)iV(10+2V-)P 

+  («-*)[V5  +  Mv5  --  2)]  X  -^5-  4cJ^ 


2. 


SURD    EViUATIONS. 


277 


)]• 

ing   complex 


?>xyz)y 

hx  -{■  ay  +  cz. 
)]  satisfies  the 

[ty,  prove  that 
iiid 

|0)2  +  <«>*z) 

'(10+2V-)T 


13.  Solve  the  Ibllowing  equations,  and  prove  that,  if  ;• 
be  a  root  of  any  one  of  them,  the  other  roots  cf  that  equa- 
tion will  be  ?•',  ?-\  r*,  etc. 


(i.)  a;^  +  l-.0. 

(ii.)  a;* +1-0. 

(iii.)  uf'-^-l-^O. 
1 


(iv.) 


(v.) 


.1' 


.3 


X  -  1 

1 


X 


0. 


0. 


(vi.) 

(vii.) 

(viii.) 

(ix.) 


X  +  1 "' 


0. 


:r  +  1 
.r«-f  1_ 


^r^  +  l 


0. 


§  61.  Surd  Equations. 
Examples. 

1.  Solve2  +  V(4'<'''-9a;  +  8)-2ar  =  0. 

Here  there  is  but  one  surd,  and  it  is  convenient  to  make 
that  surd  one  side  of  the  equation,  and  transpose  all 
the  rational  terms  to  the  other ;  this  gives 

.    V(4^''-9^  +  8)==2a;-2; 

Squaring  both  sides, 

4a;'  --  9a;  +  8  =  4a;'  -  8a;  -f  4.     /.  a;  =  4. 

2.  Solve  V(4^'  +  19)  +  V(4^'  -  19)  =  V47  +  3. 
We  have  the  identity 

(4a;' +  19)  -  (4a;' -  19)  =  38  =  47  -  9. 

Now  dividing  the  members  of  this  identity  by  those  of 
the  given  equation,  we  have 

V(4a;'  +  19)  -  -  V(4^'  - 19)  =  V47  -  3. 

Adding  this  to  the  given  equation,  then  •  . 

2V(4a;'  +  19)  =  2V47. 

.-.  4a;' +  19  =  47,  and  a;  =  ±  V*^- 


27R 


SURD    KQUATIONS. 


m 


3     V(3a;'      l)  +  V(3-a;')_« 


4 


3a;'   -  1      a-\-  b 


'd  -x' 


a 


-h 


*  *   3  -  a;"^  "    (a  -  /;)'^' 

•   ^»  =  3(a  +  6y4-(^t  -by  _  g'  +  ftZ>  +  b 
'"^       (a  +  by-l'Sia-by      a^~ab^b 


(1; 


i:^) 


4.    ?M  v(l  +  a:)  -  r?  V(l  -  ^)  ^  V(^^'  +  '^O- 
Square  both  members  and  reduce, 

Transfer  the  radical  term  and  square  both  members, 
.-.  {in^  -  n^x"  -  ^mhi^^l  -  x").  (3) 

.-.  (m'  +  w7a;''-:4mV.  (4) 

it  2mn 


x  = 


Tn?  -\-  n^ 


(5) 


The  above  follows  the  usual  mode  of  solving  equations 
involving  radicals;  viz.,  make  a  radical  term  the  right- 
hand  member;  gathering  all  the  other  terms  into  the  left- 
hand  member,  square  each  member;  repeat,  if  necessary, 
until  all  radicals  are  rationalized.  This  method  is  con- 
venient, but  it  does  not  explain  the  difficulty  that  only  one 
of  the  values  of  a;  in  (4)  satisfies  (1) ;  viz.. 


The  other  value. 


-f-  2  run 
—  ^nm 


,  satisfies  the  equation 


7n^  -f  w' 
m^{\  +  x)  +  n-y/{l  -x)=  y/(m'  -f  n'). 


SURD   E(^UAT10NS. 


279 


a) 


members, 


(2) 

(3) 
(4) 

(5) 


ring  equations 
[rm  the  right- 
into  the  left- 
[,  if  necessary, 
lethod  is  con- 
that  only  one 


equation 


The  ex^jlanation  is  simple.     Squaring  l)oth  members  ol 
(1)  is  really  equivalent  to  substituting  for  (1)  the  conjoint 
equation, 

[m^/(l  +  x)  -  nV(l  -  ^)  -  V(^^'  +  ^')] 
[^i V(l  +  '^)  +  '^VCi  -  ^)  -  V(^^*'  +  ^^0]  ^=  ^.    (<V) 
wliicli  reduces  to  (2)  above. 

Treating  (6)  or  (2)  by  transferring  and  squaring  is  equiva- 
lent to  substituting  for  it  the  equation 

[wV(l  +  ^)  -  nV(l  -  -r)  -  VK  +  ^'OJ 

X  [m^(l  f  x)  -  n V(l    -  .^)  +  V(^^'  +  ^^')] 

X  [?^i V(l  +  ^')  -f-  w  V(l     '0  -  V(^^^'  +  w')] 

X  [m V(l  +  '^')  +  ^V(l    -  ^1 4-  V(^^'  +  ^0]  -^  0,  (7) 

which  reduces  to 

[(m^  —  v})x  —  2m72-y/(l  —  a;')] 

[(771'^-  n')  a;  +  2mnV(l  -  x")]  =  0,  (8) 

which  further  reduces  to  (3). 

Thus  the  whole  process  of  solving  (1)  is  equivalent  to 
reducing  it  1y  an  equation  of  the  type  ^  =  0  and  then  mul- 
tiplying the  member  A  by  rationalizing  factors.  Thus, 
instead  of  solving  (1)  we  really  solve  (7),  that  is,  a  conjoint 
equation  equivalent  io four  disjunctive  equations.  (See  page 
191,  §  42.)  Now  the  values  given  in  (4)  will  satisfy  (7), 
the  positive  value  making  the  first  factor  vanish,  the  nega- 
tive value  making  the  third  factor  vanish,  while  no  values 
can  be  found  that  will  make  either  the  second  or  the  fourth 
factor  vanish. 

Hence,  if  one  of  such  a  set  of  disjunctive  equations  is 
proposed  for  solution,  the  conjoint  equation  must  be  solved ; 
and  if  there  be  a  value  of  x  which  satisfies  the  particular 
equation  proposed,  that  value  must  be  retained  and  the 
others  rejected. 


i2S() 


Sl'Kh    KlilATIoNS. 


This  pn»(;('ss  is  the  o|ijM)Mi(i'  to  thiit  given  in  §§  412  an<l  43  ; 
tliere  ii  con  joint  oquation  is  solved  by  resolving  it  into  its 
equivident  disjunotivo  equations.  The  two  jn-ooosses  are 
rehited  somowliat  as  involution  an<l  evolution  are. 

Further,  it  should  1)0  noticed  that  just  as  there  are  four 
factors  in  (7)  wliile  there  are  only  two  values  in  (4),  it  will 
in  general  be  possii)lo  to  form  more  disjunctive  equations 
than  there  are  vabies  of  x  that  satisfy  the  conjoint  equation, 
and  consequently  it  will  be  possible  to  sele(;t  disjunctive 
equations  that  are  not  satisfied  by  any  value  of  x,  or,  in 
other  words,  whoso  solution  is  impossible. 

This  will  perhaps  be  better  understood  by  considering 
the  following  problem  : 

Find  a  number  such  that,  if  it  be  increased  by  4  and  also 
diminished  by  4,  the  difTerence  of  the  square  roots  of  the 
results  shall  be  4. 

Reduced  to  an  equation,  this  is 

V(^  +  4)  -  V(^  -  4)  =  4.  .  (0) 

Rationalizing,  this  becomes 

[4-V(^-  +  4)-f-V(^-4)]  ^ 

x[4-V(^+4)-V(^-4)]  •      • 

■    x[4+V(^+4)  +  V(^'-4)] 
X  [4+ VC-^' +4)  -  VC^-4)]  =  0, 

which  reduces  to 

[24 -^  8  V(^  +  4)]  [24  +  8  V(^  +  4)]  =  0  ; 
that  is,  9  —  (a;  -f  4)  —  0,  or  a;  =  5. 

Now  X  =^b  satisfies  (10)  because  it  makes  the  factor 

4-V('^  +  4)-V(^--4) 

vanish,  and  it  is  the  only  finite  vahie  of  x  that  does  satisfy 
(10),  or,  in  other  words,  there  are  no  values  of  x  which  will 
make  any  of  the  factors 


(10) 


42  and  43  : 
r  it  into  ilrt 
irocosses  arc 
,re. 

oro  are  four 
n  (4),  it  will 
vo  cq\iations 
)int  equation, 
t  disjunctive 
3  of  X,  or,  in 

^  considering 

by  4  and  also 
3  roots  of  the 


(^) 


(10) 


0; 


the  factor 


SURD   EQUATIONS. 


281 


lai  does  satisfy 
)f  X  which  will 


or 


4-f  V(-^-f-4)-V(^-4), 


vanish.     There  is,  therefore,  no  numl)er  that  will  .satisfy  the 
conditions  of  the  problem. 

It  will  bo  found  that  as  x  increases  ■\/{x-\-^)  y/{x  A) 
decreases;  hence,  as  4  is  the  least  vaUe  tliat  can  b(»  given 
to  X  without  involving  the  s(|uan'  root  of  a  negative,  the 
greatest  real  value  of  y/ix-}-^)  -  y/{x  A)  is  -y/S,  whicli 
is  less  than  4,  We  sec  by  this  that  our  method  of  solution 
fails  for  (9)  simply  because  (9)  is  impossible. 


5.    v[(^*  4-  x)  {b  -h  x)]  -  v[(«  -  ^)  (^  -  ^')] 


(1) 


Collecting  the  terms  involving  ^(a  +  x)  and  y/(a  —  x), 
respectively  the  equation  becomes 


[^(a-\-x)-y/(a-^x)]  [  y/{h\x)-\-y/(b 
This  is  satisfied  if  either 

or  V(^  +  x)  +  V(^  -x)  =  0. 


x)]-=0.      (2) 

(3) 
(4) 


The  rational  form  of  (3)  is  (a-\- x) ~ (a  —  x)  =  0,  which 
is  satisfied  by  x  =  0,  and  this  also  satisfies  (3). 

The  rational  form  of  (4)  is  (b  -{-x)  —  (b  —  x)  =  0,  which 
requires  a;  =  0  ;  but  this  does  not  satisfy  (4).  Hence, 
the  second  factor  of  the  left-hand  member  of  (2)  can- 
not vanish. 

Therefore,  the  only  solution  of  (2),  and  hence  of  (1),  is 
a:  =  0,  derived  from  (3). 


282 


SURD   EQUATIONS. 


6. 


7. 


</(«  +  x)+  -^(a  -  x)  =  -i/{2a).  . 

Cube  by  the  formula, 

(u  +  vy  =  w'  -f  v^  -{-Suv(u-\~  v). 

,',  (a  +  x)  +  (a  -x)-^S  ^[2(1  (a'  -  x^)]  =  2a. 

.\2a(a'-x')  =  0. 

.\x  =  ±a. 

Both  these  values  belong  to  the  proposed  equation. 

The  rationalizing  factovs  of 

■v/(«  +  a:)  4-  ^(a-x)  -  ^(2a)  =  0 

are      -{/(a  +  .-r)  +  w  ^(«  —  x)  —  w^  -^(2  a), 

and     -^(a  -\-x)-i-o)^  -{/{a  —  x)  —  co  -y/(2  a). 

See  Exam.  11,  page  276. 

The  remarks  on  Exam.  4  will  apply  mutatis  7nutandis 
to  equations  of  this  type. 


^(a  +  xf  +  ^(a'  -  x')  +  ^(g  -  >r)^  _ 

Assume  -^(a  {-x)  =  u  and  -^(a  —  x)  —  v. 
: _u^  -{-v^  =  2a  and  u?  -v^  =  2x, 


(1) 


8. 


w 


v 

3     I     „^i 


X 

a 


u  -\-  V 

Also  (1)  becomes 

u^  -\-uv-\-  V 
u^  -{-uv-{-  V 


(2) 


;,  =  c- 


(3) 


Multiply  both  numbers  by 


w'  -  v' 


u  —  V 


v^  -\-v^         ?/  4-  V 


SURD   EQUATIONS. 


283 


)]  =  2a. 


^uation. 

0 
). 

itis  mutandis 

(1) 


(2) 


(3) 


•••  I'y  (2), 


X  U—  V 

-  =  C -—• 

a        u-f-v 


(4) 


Again,  adding  and  subtracting  denominators  and  numer- 
ators in  (3), 

uv         c—  1 

Adding  and  subtracting  2  (denominators)  and  numera- 
tors in  this, 

u^  -—  2uv  -\-  v^  —  _^^^ 


or 


fu  —  v" 

\u-\-Vj 


3  c?  -  1 

.*.  substituting  by  (4), 
^^^2  3  -c 


a 


3e-l 


8.    [■^{x-^a)^--^{x-a)f[-^{x-\-a)--^{x-a)]  =  2c.  (1) 

Assume  w  =  ^(a;  +  a)  and  i*  = -v/C'^  ~"  ^)> 
and  (1)  becomes 

{u-{-v)\u  —  v)  —  2c, 

or      (u  +  v)\u'^  —  v^)  =  2  c. 

Also      w*--'y*  =  2a 

or      (u'-{-v'')(u''-v')  =  2a, 

and    u*-]-v*  =  2x. 

From  (2)  and  (3), 

(u-vy{u''-v')  =  ^a-~2c.  (5) 


(2) 

(3) 
(4) 


284 


SURD   EQUATIONS. 


(^>) 


••■  (2)  X  (5), 

{v^  -  vJiv?  -  vj 
or       (y?  —  v'^y  =  ^c(2a  —  c). 
Also  (3)'  +  (6), 

=  4(a'  +  2ac-c2) 
or       (w*  +  v')  {y?  -  'u'f  =  2{a^  +  2ac- 
Substituting  by  (4)  and  (6), 

2x  ^{2ac  -  c")  ==a?-\-  2ac  -  c\ 


c% 


Divide  the  terms  of  the  identity 

■i/{a  +  xy  -  -^{a  —  xy  =  2  x 

by  the  corresponding  terms  of  the  equation. 


\\a-x)     c-l 

mi- 


.  o,-{-x 


a  —  x 


x 


{c  +  1)*  +  (^  -  1)* 


10.    ^{a  -  xj  +  -{/[(a  -  ^)  (i  -  ^)]  +  ^(^  -  xf 
=  ■^(aJ'  4-  a6  +  h'). 

Divide  the  terms  of  the  identity 

■^(a  -  xf  --^{h-\-xJ  =  a-b 

by  the  corresponding  terms  of  the  equation. 

a  —  h 


:.-^{a-x)-y{h-x)  = 


^{a'  f  ab  +  6^) 


SURD   EQUATIONS. 


285 


(f>) 


c^). 


-x)]  =  2cx. 


ition. 


■  xy 


ition. 

b 

ib  +  bi 


Cube,  using  the  form 

(w  —  v)'  =  m'  —  v'  —  3  uv  (u  —  v). 
(a  —  x)  —  {b  —  x) 


11. 


a  —  b 


-Z-^\(a-x)(b-x)'\- 

^_(a^^hl_^^_^_Sab(a^b)^ 
a'  +  ab-^b'  »        .      .- 

,:-^[(a-x){b-x)]  = 
.'.  (a  ~  x)(b  —  x)  = 


a"  -\-  ab  +  />' 

ab 

^(d'  +  ab  +  by 

a'b' 


{a'  -i-ab-\-  bj 
a  form  solved  in  (0),  page  231. 

(a  -  xf  V(a  -x)-{~(x-  by^jx  -  b)  _  ^     ^ 
{a  —  x)^{a  —  x)-\-{x  —  b)-y/{x  —  b) 

Write  a  —  i  in  the  form  {a~  x)-\-{x  —  b),  and  multi- 
ply by  the  denominator  of  the  left-hand  member. 

.-.  {a-xY^{a-x)-\-{x-bY^{x-b) 
=  (a-xy-^{a-x)i-(x-by^(x-b) 
+  (a-x)(x-b)[^(a-x)+  ^(x-b)l 

.-.  (a  -  x) {x  -  b)[^(a  -  x) -}-  y/(x  -  b)]  =  0 

.*.  (a  —  x)  —  0,   or  X  —  b  ~  Oy 
or     -y/  (a  —  x)  -{■  -^/{x  —  b)  =  0. 

X\     Cly  U/J     O, 

The  equation  -^(a  —  x) -}-  ■y/(x  ~  b)  =  0  has  no  solu- 
tion, for  the  sum  of  two  positive  square  roots  cannot 
vanish. 

The  solution  x=  h{a-\-  b)  belongs  to  the  equation 
■y/(a-x)-'^{x  -b)  =  0. 


286 


SURD   EQUATIONS. 


2.      a/a  — 3;   ,      :3  +  3: 
\b-\-x      ya—x 


—  c. 


Square   both   members,  subtract  4,  and    extract  the 
square  root. 

Ihi-x_ 


a; 


a  —  a: 


i  [c  i  VC'^' 


V(^'  -  4). 
4)]  =  e,  say. 


=  e\ 


5  +  a; 

.  2x-(a-b)_l-e' 
a-\-h  1  +  / 


•      •     •V 


\ 


{a-b)-{-{a-^h) 
Or  thus,  cube  both  members. 


l-\-^ 


]■ 


a  —  X 


X 


+  3«?4- 


^>  +  a:_ 


a  — a: 


6  + 


(a  —  a:)  (i  -f-  ^) 


C 


(^+^-)-( 


a 


3c-2 


(h  +  a;)  +  (« 

_  (c  +  iy(c  -  2) 
{c~\)\c-\-2) 

2x-{a-h)  _c^l 


c'-3c+2 


a 


+  h 


\c-2 
\o  +  2 


c-iylc  +  2 


Prove  that 


l-\-^     c 


4 


2 


\\c+2 


if     2e  =  c±V(^''-4). 


SURD   EQUATIONS. 


287 


1. 
2. 
3. 
4. 
5. 
6. 

7. 
8. 

9. 

10. 
11. 
12. 

13. 
14. 

15. 

16. 
17. 

18. 


Ex.  67. 

V(a:  +  4)  +  V(^-3)  =  7. 

V(3a;+l)-hV(4:i'  +  4)-l. 
V(2a;  +  10)  +  V(2^  -  2)  =  6. 
■y/(pnx)  —  -\J{nx)  —  7n  —  n. 
■y/(bx)  4-  ^(ab  -f  bx)  =  y/x, 

V.  +  v(.  +  3)=— ^^. 
V(«^  +  ^')  =  (i  +  ^)- 

^(17:. -26)  =  I 
■y/x  -  V(a  +  a:)  =  J-. 

^  +  ^  -  V(^'  +  ^'0  =  ^'• 
V(8  +  x)-^x=^  2  V(l  +  ^). 
V(2a:  -  27a)  =  9  Va  -  V(2^). 

5 V(2.r  -  1)  +  2V(3.r  -  3)  _  p^ . 
4 V(2:r  -  1)  -  2V(3:r  -  3)      ^*' 

V2a;+ V(3-2a:)  ^  3 


V2 


.r 


'(3  -  2  a;)      2 


2^(3a:  +  3)  +  -^(7a-  +  8)_g 
2^(3a;  +  3)--</(7a;  +  8) 

33  [13  -  2  V(^  -  5)]  -  3  [13  +  2^{x  -  5)]. 

(V^  +  1)[ V(^^  +  1) -  V^^] 
^--  (  Vn  -  1)  [  V(^^  +  1)  +  V^i^]- 

V(a;  +  g)  +  V^    _  ya:  +  Va 


V(^  +  c)  -  v*     v^  ~  V 


a 


288 


SURD   EQUATIONS. 


19     Va:  +  28  __  Va:  +  38 

V^  +  4  "    v^  +  ^" 

-</2a;  +  9       ^2a;+15 

21  VaM-2a___  Va?H-4a 

22  32^-1    __1  +  V3a; 
V3^  +  l  2 

23.  ^H-Vi^Zl^^-a. 

24.  V^  +  V^  —  ^ 
+  1  +  V(aV  -  1)  _.  ^^ 


25. 


26. 


27. 


ax 


ax 


+  1-V( 


2     '2 

ax 


1) 


1^^ 


i+v 


1-^(1-^) 


[i-V(i-^) 


J==a. 


-</(a:  -  1)  +  ^{x  -  1)      2 


28.  ^(1  -  a:)  +  ^(1  +  a:)  = -{/3. 

29.  -{/(3  +  rp)  +  ^/(3  -  ar)  =  ^7. 

30.  ^(a;+l)-^(^-l)  =  ^/ll. 

31.  ■^{a^x)-\-Aj{a-x)=^^h. 

32.  ^(1  +  V^)  +  ^(1-V^)^2. 

33.  ^x  —  V[a  —  V(«^  +  ^')]  =  i  V 

34.  -^(25  +  a:)  +  ^/(25  -  a;)  =  2. 


35.    a;+V('^'  +  ^') 


wa 


V(«'  +  ^') 


SURD    EQUATIONS. 


280 


36. 
37. 

3d. 
39. 

40. 

41. 
42. 
43. 
44. 
45. 
46. 
47. 

48. 
49. 
50. 

51. 

52. 

53. 

54. 


V(i  -}-  x)  -f  v[H-  :r  +  v(i  -  ^0]  =  V(l  -  -  ^). 

V(l  +  ^^'  +  ^'')  +  V(l  - ^  +  a;')  =  ^>i^. 

bx  —  c'    _  V(6.r)  +  c  __  ^ 

^(2x'  +  5)  +  V(2^'  -  5)  -  Vl^  -1-  V^- 
V(3:r'+ 10)+ V(3^'- 10)  -  Vl'^  +  V37. 
V(3^'  +  9)  -  V(3^'  -  9)  -  V34  +  4. 
V(3a-  36  +  a:')  +  V(2a  -  2^>  4-a;')  --  V«+  V^- 
V(4a'  -  36=^  -  2a:2)  +  ^(Sa"  -  36^  -  a;'^)  -=a+  a;. 

[  V(«  +  ^)  +  V(«  -  ^)][  V«  +  V(«'  +  ^)]  =  2a;. 
V(a;'  +  2ax)  +  V(^''  -  ^(^^)  = 


waa; 


V(^  +  2aa:) 


-a'). 


V[(2a  +  a:)^  +  ^'^J  +  V[(2«  -  a;)^  +  b']  =  2 


a. 


rV(a-:r)+V(a:-^>)r 


V(a  —  ^)  -  V(^  —  ^) 


=  V^- 


V(l+a;')  +  V(l-a;^),   a 

V(i  +  ^^)-V(i-^"0    * 

^(l+a:',+-^(l-a:^)_a 
^(1  +  x')  -  ^(1  -  x')      b 

</(l-^x')  +  -i/(l-x^)_a 
^(1  +  x')  -  </(l  -  a;')      b 

■i/(l-}-x')+-i/(l-x')__a 
</(!  +  ^')  -  -^(1  -  ^')      b 


290 


SUKD    EtiUATIONS. 


65. 

56. 

57. 
58. 
59. 
60. 
61. 
62. 
63. 
64. 
65. 
66. 
67. 
68. 
69. 
70. 

71. 
72. 
73. 

74. 


2a:)  =  2  V 


^(1  +  x')  -  ^:y  -  1)  "  b 

V(.^  -f  1)  -  -Vi^'  -  1)      h 

V(4a  +  b  -  4a:)  -  2  ^(a  +  b   -  2x)  -  ^b 

V(3a  -  2^  -I-  2x)  ■  -  V(3«  -  2b 

V(2a  -  ^»  +  2a:)  -  -yjiy^a  -  96  -  6a:)  --  4  V(«  -  ^)- 

V(3a  -  4 Z>  +  5a:)  -I-  VC^'  " "  «)  =  2  V(a:  +  a). 

V(3a  --  46  -f  5a:)  +  VG^•  -  «)  =  2  V(2a:  -  26). 

V(5  a:  -  3  a  +  4  6)  +  V(5  .r-3a-46)  =  2  V(^'  +  «) 

V(2a+64-2a:)-(- V(10a+96-6a:)  =  2V(2a+6-2a:). 

2V(2ai-6+2a:)+ V(10a+6-6r)  =  V(10«+96~6.'>:). 

V(2a-136-hl4a:)+ V[3(6-2a+2a:)]  =  2V(2a-6+2a:) 

V[3(7a  +  6  +  ^)]-V'(«+76-a:)  =  2V(7«-f6-a:) 

V[(«  +  ^)(a:  +  6)]  +  V[(«  -  ^)(^  -  ^)]  --  2^/(o,.r). 

V[(a  +  x){x  4-  6)]  -  V[(a  -  ^)(^  -  *)]  =  2  V(^^). 

^{ax  -f  a:')  ~  V(^^  ■"  ^')  —  V(2«^  —  a^). 

y/{ax  —  3?)  +  V(<^^  +  ^^)  —  V(2«^'  +  «^)- 

L__  + L__ 

i  +  va-^)   i-v(i-^) 

X  +  ^{ax)      a  +  V(aa:)  __  ^^ 
a  —  ^{ax)      X  —  ^{ax)         a 

^\{a-^x){x-\-b)-\  +  y/[{a-x){x-~b)'\  _     \a 

V[(«  +  ^0  (^-  +  ^)]  -  V[(«  -  ^)  (^  -  ^)]     \^ 


—  2 


a 


a:. 


4 


Sa~2b  +  2x ^  [Va  +  V(2a-26)1' 
3«-26-2a:  26-a 


SURD   EQUATIONS. 


201 


2  V'"- 

[  V(«  -  ^)- 

a). 
-2h). 

}-b-2x). 

/(2a-b-\-2x). 
1a-\-h-x). 

2  V(^^)- 


.4 


75.  ■^{a~\-x)-\--^(a-x)  =  2'^a. 

76.  ^{a  +  xy  -  </(«'  -  x')  +  </(a  -  xf  =  ^a». 

78.  ^(l+^)'  +  </(l-^)'  =  2i-^(l--:f'). 

79.  ^(3  +  :z^)  +  ^(3  -  ^)  -  ^6. 

80.  ^(1  f  a;y  +  ^(1  -  xf  -  5  [sy(l  +  ^)  -f  ^(1  -  x)]\ 

81.  ^(14  +  a:)' -  ^'(196  -  a;2)  + -^(14 -~  a:)^  -  7. 

82.  [^(9  +  a;)  +  ^(9-a:)]^(81-a:'')=12. 

83.  [-^{Ui-xy-^{u-xy][-^(u-\-x)~^(U-x)]  -  le. 

84.  [^{b7+xy+^{b7-xyi-^ib7-x)+^{bnx)]  =  100. 

85.  5[^(41+:r)+^(41-.r)]^  -  8[  V(41+:r)+V(41-^)]. 

86.  [-</(:r  +  5)4-</(^-5)7[</(a;  +  5)--</(a;-5)]  =  2. 

87.  [^(x+  l)  +  ^(:r-  1)][V(^+  1)  +  V(^-1)] 

=--26[^(:r+l)-</(a:-l)]. 

3/1  —a;  _ 


-  <It^Mi 


+  x 


O"   iy-\-y  ^  =  a). 


89.  2[^{\-\-xy^-^(l-x')]  -  (o'^+l)[^(l+:^)+^^(l"^')r- 

90.  ■^{a-^x)-\-^{a-x)==yc. 

91.  [^(a  +  :^)  +  ^/(a-a;)]S/(«'-^')==c. 

92.  ^(a  +  xy  =  ^(a'  ~  ^')  +  ^(«  -  ^)'  =  ^c^ 

93.  [-{/(a  +  xy  - ^{a  -  xy][-^{a  +  x)  -  ^{a  -  x)\  =  c. 

94.  [^(a  +  xy  +  </(a  -  xy]  [-^(a  +  a;)  +  ■</(«  -  x)]  -  c. 

95.  (a  -|-  x)  -y/{a  —  x)  —  (a  —  x)  -^{a  +  x) 

=  c[-^(a -\- x)  -  ^(a  -  x)]. 


SURD    EQUATIONS. 


96.  {'I  -1  .r)  ^{a  +  x)      {a      x)  -^{a  --  a-) 

97.  [-</Gx  +  aO'-\/K-^-')-f  \/(^~^)? 

^c[y{a  +  x)  \--^(a-x)l 

98.  [</(a+.r)^-</(a-.^•)7■=(c^-l)[V(«+:r)+V(«  -^)]- 

99.  [^{x  +  «)  -  ^(^^  -  «)][  V(^  +  «)  +  V(^  -  «)P 

=  ^[</(.2;4-a)  +  ^(a:-a)l. 

00.    V(^  --  (^)  +  V(^'  -  ^)  _     l£_i_«. 


01. 


02. 


03. 


04. 


05. 


y/(x  —  a)  —  ^{x  —  b) 

V(a;  —  g)  +  ^{x  —  b)_     ja  —  x 
^{x  —  a)  —  V(^  ~  ^) 


\^ 


\a  —  a;  _     1 

3  [ft    —  X  3  I 

Sli  +  r.      V 


=  c. 


a  —  X 


X 


=  c. 


a  —  X 


h  -I-  a;  __ 
a a; 


106. 


107. 


108. 


t 

Jo— ^  _    4I: 
\h-x      V 


a  —  rr' 


a  —  a; 


6  +  a; 
a  —  X 


-_x 
a  —  x 


=  c.      109. 


110. 


X 


=  c. 


Ill, 


a—x 


ylb-x'"\ 

gla— £ 


a: 


a  — a; 


b  +  x 


=  c. 


X 


\a- 


=  c. 


X 


112. 


113. 


^(a-^y+^(b-xy_^^ 
■y/((X  —  a:)  +  V(6  —  a;) 

V(a  -  a)'  +  V(^>  -  '^y  ^ 
W(a-x)+^ib-x)J    ' 


SURD    EQrATIONS. 


293 


i 


h  —  X 

a  —  x 


=  c. 


\h^-x__ 


\a  —  x 


114. 


115. 


116. 


117. 


rV(a-:r)4-V(^>-^)?_, 
V(a  -  a;)  —  V(^  ~"  ^) 

-^((t  -  a;)  4-  V(a;  —  h) 
[V(«-^)-V(^"^)r 


c. 


"-  ^::g:^;::g-v[(^-^(x+^)]. 


119. 


120. 


v(«  -  xY + v(^ + hf  _       (« + ^y 


V(a  -  ^)  +  V(^  +  *)      4  V[(«  -  x)  {x^-b)] 

x^  +  (g  —  a:')  ^{a  —  a:')  _ 
a;4- -^(a— a;*) 

121.  ^^ + («'  -  ^;)y(«;  -  ^ = ,^  v(«'  --  ^■')- 

X  -\-  y/{a^  —  x') 
122.    -^{a  - a,f  -  ^[{a  -x){x-  b)]  +  ^^/(ar  +  bf 


123. 


124. 


125. 


126. 


=  -^{d'  -  ab  +  b'). 

b  V(c^  -  x)  -]-  a  ^{x  —  b)  _ 
^(a-x)-i-^(x-~b) 

a  V(«  —  x)  4-  b  V(.-r  ~b)  _ 
^{a  -  a;)  +  V(a;  —  ^) 


.r. 


X. 


V(.r  -  g)  +  V(a-  +  a)  —  V(2a)  _    4/a:  +  c 


i 


X 


■y/(x  —  a)  —  ^{x -\- a) -i- -y/{2,a)       \x—c 

v(a  -  a-)  4-  yg  ^ 

V(^  -  ^)  +  V<^ 

127.    y >  -  a;)'  -  ^[(a  -  x)  (x  +  i)]  +  -^(x  +  bf 
=  -^(a'-ab  +  b'). 


~\a;- 


294 


SURI>    KQITATIONS. 


128.  ];/ia      xY      ^[{a-x)(x      h)]  \  ^{x      hy\' 

=  (a--/>)[sy(^e      x)  \  \/(x  -  />)|. 

129.  \^(a    xf'ty/i^^+'^^yf-   (<tf/>)|\/(«    .Of\/(^>KOj- 

130.  ^(a  -  x)  -H  y/U'      b)  -  ^c. 

131.  ^{a  \-  xf  -  ^(a  -  xY  =  ^{2cx). 

132.  sy(/i  -  .^0'   f-  ^/[{a  -  -  .r)  (7;    -  x)]  -\-  ^{h      xf  -  -^c\ 

133.  ^{a   -  xf  -  ^[(a  -  -  x) (x  -^  b)\  f  ^{x  +  ^»)' 

134.  [ </(n  -  .r)  -f  ^{x  +  />)l^r(^«  -  -'^•^  (^  +  ^0]  -  ^.      ■• 
136.  -^{ii  -  xy  -}-  ^(x  --  <{>J^  ==  c  [-^(a  -  x)  -f  ^/ix  -  -  A)]^ 


136.    0:4-  ^/(a''-z'')  = 
137. 


a 


_x-{-^{^2b^-x^) 


x'~b       x~^(2b'-x') 

138.  (a  +  a:)  ^(«  +  5;)  4- (a  -  a;)  ^Ca  -  a;) 

=  a[^(a  +  x)-{-^(a-x)]. 

139.  (a  4-  ^^')  </(«  -  a;)  4"  («  -  a:)  </(«  4"  ^) 

-  «[^(a  4-  4  4-  </(«  -  x)]. 

140.  </(26  -  x)  4-  v/(a:  -  10)  -  2. 

141.  [</(a-a,-)4-</(^-^')r-=^[V(«--^)  +  v('^--^)]- 

142.  {a-x)^(a-x)  +  (x~-b)-^(x-b) 

=  (a  --^>)[^(a  -  x)  4-  </(a;  -  b)]. 

143.  [</(«  -  .r)  -f  ^(x-b)J[-^(a-x)  4-  V(^-  <^)] 

=  c(a-^b  -2x). 

144.  [^'(a  -  ar)  4-  W  "  ^)]  [  V(«  "  ^)  +  V(^  ~  ^)]' 

145.  a-,y{l  +  x^)-Xy/(x^-{-a'')^e. 


\ 


SURD    KQUATIONS. 


295 


)f  </(/>i^)j- 


b)]--c.      • 


;)  +  yi'^  -  ^)]- 


146.  {a  -x)'^(x~b)-\-(x  —  b)^(a-x) 

=  c[-^{a--x)  +  ^(x~b)]\ 

147.  [ S^/(a ~x)  +  -^(b  +  :t')r  -  o[-^{a  -  x)^  -|-  sy(/>  +  ^)']. 

148.  [  ^(a  -  x)  4-  </(i  +  x)]^  -  c  ^[(a  -  x)  {b  +  a--)]. 

149.  ■{/(«  -  :?;)■'  -  •</(6  -  a:)'  =  csl/(a  -f  6  -  2a;). 

150.  -^{a  -  x)  +  ^(x  —  b)  ■-= -{/c. 

151.  v^(a  -  a:)  +  ^(o,- —  i)  ^  ^c.  . 

152  (^^  -  ^)  </(«  -  ^0  +  C'^-  -  ^)  </('^'  -  ^) 
(a  -  a;)  <7(a;  -  ^>)  +  (:c  +  b)  ^{a  -  a:) 

153  i^-^)  W'  -  ^)  +  (^  -  x)^/(a  -  x) 

'^(a  -x)-  -^(b  -  x) 

154.    ^(«-^)  +  -v/C^^-   ^)_  0 


=  c. 


=  c. 


■^(a  -x)-  ^{x  -b)     a  H-  6  -  2a; 

155  [</(«- ^)+-t/(^-^)r„, 

•       ^(«-a;)-</(6-:i-)  ' 

156.  (a  — a,•)^/(rt  — a;)-(a:  — ^) -^(a:  — ^>) 

=  c[-^{a-x)--<:/{x-b)], 

157.  (a  -  a:) -^(a;  +  ^>)  -  (a;  +  ^)  ^(a:  -  a) 

=  c[^(a-a;)-.^(:i'  +  i)]. 

158.  [^/{a  -  xY  +  ^(a:  -  bf]-^[{a  -  x){x  -  b)]  =  c. 

159.  [-^'(«~ ^) - a/(^ - ^)r[A/(^^ - ^y- vi^-^y] = ^. 

160.  [^{a-  xy-  ■^{x-by][y/{a-  x)-\-  -::/{x  ~b)]-^  c. 

161.  [^(a  -  a;y  +  ^{x -f  5^7  =  c[^(a  -  .r)  +  ^(^'  +  6)]. 

162.  V(^'-«'-^')  +  V(^'-^'-^')-V(^'-^'-a')  =  ^- 


163. 


7?l' 


x'' . .  a/0>^'^-  -I-  2)  -{-yjjm'x  -~  2) 


X 


a'        V(^'-^  +  2)  --  ^{m'x  -  2) 


206 


SURD   EQLTATIONR. 


165.    [ ^(a  -  :r)  +  -^(6  -  x)'][ </(a  -  a;)  -  ^(6  -  x)]  =  c. 

166     v^C*^  —  a:)  —  -y/jx -  -  &)  ^ a-\-h  —  2x 
■{/{a-x)-\-y{x~by        a-h 

167.    ^(a  +  a;)+-^(a-a;)=-^(2a). 

Write  w  for  -y/(a  —  a;),  and  v  for  -yj{:c  ~  h). 
169.    y(5-3a;4-:f'0  + V(5-3y  +  3/^)-6;  a:  +  3/ 

170.  v(^  — ^y)  + V(y-^2/)  =  «;  ^  +  2/  =  ^- 

171.  i/{x^m)  +  y/{y-\-n)  =  a;   x  +  7j-=b. 

172.  -^(143  +  0,-)- ^(y- 18)  =  1;    x-y^bO. 

173.  v(^'y)  +  V[(i-^)(i--y)]=^«; 

174.    xy  +  ^[{\-x'){\-f)-]  =  ab- 


=  3. 


^va-y')+2/V(i-'^*0  = 


a' 


2a6 


175.  v(^-^y)  + V(y-^2/)  =  «; 
v'(^'  -  ^')  +  v'(y  -  2/')  =  ^• 

176.    :ii  +  3/S-=3(:ri-?/3)-^3a:. 


177. 


a 


a  ~  x" 


^\h.  fa 

\y 


\y'  -  ^ 

xy  —  ah. 
178.   (^  +  y)*H-(;-y)i-a4; 


''  -f-  6'^  ^  cfc-^  +  x'J 


X. 

(b-x)-\  =  c. 


X  —  h). 

=  b. 

b. 

-=50. 


=  3. 


Ai 


=  4; 


CHAPTER   IX. 

Cubic  and  Quartic  Equations. 

§  52.    77ie  Oubic.    Let  the  general  cubic  equation  be 

ax"  +  Sba;^  +  Scx-\-d  =  0.  (1) 

Let  y  =  ax  -\-  b.  (2) 

Substitute  •— 


a 


for  X  in  (1),  and  multiply  the  resulting 


equation  by  a}, 

2/^  +  3(ae  -  &^)y  4-  {a\l -  Said?  +  2b^)  =  0, 
which  may  be  written 

if^-^Hy  +  0^-=r.O^  .         (3) 

in  which          //=  ac  —  b^,  (4) 

and                   G=a''d-3abc  +  2b\  (5) 

Assume        y  =  w  -y/ri  +  w'"*  \/^''2,*  (6) 


in  which 


0) 


(V) 


Cube  (6),     3/*  =  r,  +  r,  +  3  -^(nr.Xio  ^r,  +  o>^  ^r,) 
=  n  +  ^2  +  3-^(r,r,)?/; 

2^'--3^(r,r,)y-(n  +  rO  =  0.  (8) 

Equate  coefficients  in  (3)  and  (8), 

n  +  n^-G,  (9) 

and  -^(r,r,)  =  -ir.     .\nr,  =  -lP.  (10) 

*  Throughout  this  chapter  the  symbols  y/  and  y/  will  be  used  to 
denote  the  corresponding  arithmetical  roots  of  the  quantities  they 
,  operate  on.    General  roots  will  be  denoted  by  exponents.     See  §  50, 
[page  264. 


298 


CUBIC    AND    QUARTIC    EQUATIONS. 


(9)  and   (10)  show  that  Vi  and  o'z  are  the  roots  of  the 
quadratic 

r'-i-Gr--IP  =  0,  (11) 

which  may  be  written 

(2r  +  Gy  -  (G'  +  4^^)  =  0.  •  (12) 

By  (4)  and  (5), 

G^-{-^II'  =  a'  (a'd'+ 4ac='  -  Qabcdi-  Wd  ~  Sb'c') 
=  a'\BSLy.  (13) 

.\{2ri-Gy-a'A  =  0.  (14) 

.-.  r  —  —  ^G±:  ^a^A. 
Let      n  =  -hG-ia-y//^.  (15) 

.'.r,  =  ~W+la-^A.  (16) 

Substitute  these  values  for  ri  and  o'i  in  (6). 

.-.  y  ■=  o,^(-i6^-^aVA)+o>^^(-^G^+^aV^) 

==-u>^C2G-^la^/^)-io'-^(^G-ia^A).     (17) 

Substitute  this  value  of  y  in  (2). 


-o>^-</(^(?-iaV^)]. 
Hence,  if 


(18) 


x,^^[-h--^{hG-{-lay/^)~y{\G-\ay/^)l 

.r,  =  i[-Z,4-  i(l  +  2  V3)-^(J  6^+  1«V^) 

+  Kl  -  ^"  V3)a/(2^  G  -  la^^)l         (19) 


(fc 


1 


0:3  =  i[- 5  +  1(1  -  z  V3)-^(^  (7+ ^«V^) 

^-l(l  +  ^•V3)^(*6'-daV^)]; 
in  which  a'^A  =  6^^  +  4  .ZP, 

II=ac~h\ 


(20) 


CUBIC   AND   QUARTIC   EQUATIONS. 


299 


roots  of  the 

(11) 

•  (12) 

(13) 
(14) 

(15) 
(IC) 


(18) 


'^)l      (19) 
(20) 


If  the  cubic  have  one  or  more  rational  linear  factors,  the 
above  method  of  solution  should  not  be  attempted ;  but 
such  factor  or  factors  should  be  determined,  and  the  cubic 
resolved  into  its  equivalent  disjunctive  equations,  and  these 
solved.  (See  Note,  page  233.)  If  the  cubic  have  no  rational 
linear  factor,  -y/^i  G  -\-  ia^A)  will  not  he  reducible  to 
the  form  u  +  V^  5  ^^^'  ^^  ^^®  cubic  have  such  factor, 
■\/{lG -{■  ha^A)  will  be  reducible,  and  the  reduction  may 
he  effected  hy  resolving  the  cubic  into  its  equivalent  dis- 
junctive equations,  and  solving  these. 

Examples. 

1.    Solve  9.r'-9a;-4-:0. 

Assume      x  =  o)  -y/ri  -f  a^  ■y/r2. 

,-,o^~ri-\-  9-2  +  3  ■y/{ryr.^  x, 
.'.  n  +  ^'2  =  -|, 


and         y/inr^)  =  i- 

.-.  n  = -J  and  r.,  =  f 

2.    Solve  2a;''  +  6a;'+ 1  =  0. 

First  Solution. 

Assume     y  =  x-\-\,  and  substitute  for  x, 

.•.23/»-6y  +  5  =  0. 

Assume     3/  =  to  -^r^  +  w'  -y/r^, 

cube  each  side,  and  compare  coefficients  with  those 
of  the  equation  in  y, 

n  +  ^'2  =  -|; 


300 


CUBIC   AND   QUARTIG   EQUATIONS. 


and 


.-.  r'  +  |r+l=-0. 
.-.  n  =  -  2, 
^2  =  -  i- 

.•.a:,  =  -l--^2- 


-^2 


Second  Soluiio'n. 

Let  21  —  a:~\  and  substitute  for  x, 

.-.  0'  +  62  +  2=-O 
Assume      z  =  (a  -y/r,  +  o>^  -^rj. 

.'•  n  +  r^  =  -  2 ; 

.*.  n  =  -  4, 

r^  =  2. 

1 


r,r. 


and 


x.= 


^2 --^4 


_^4  +  </8  +  ^16 
2-4 


3.    Solve         a7  +  y'  =  7; 


a;*-22a:^  +  :r  +  114  =  0. 

(x  -  3) (.r''  +  3a;^  -  13y  -  38)  =  0. 


0. 


CUBIC   AND   QUARTIC   EQUATIONS. 


301 


Therefore,  either 

a:-3  =  0, 

or  else    a:'  + 3:r' -  13.r- 38  =^0. 

Let  2  =  :r  +  1, 

and  substitute  for  x  in  the  latter  of  these  disjunctive 
equations. 

.•.2'-16z-23  =  0. 
Assume     z  =  w  -y/ri  -f  oy^  ^r,. 
.•.n  + 7-2-23, 

.-.  r  =  -J  [23  ±  V(23'  -  -^^f*)] 
=  i(23d=-JV6303). 

.-.37=3 

- 1 4-  <o  ^/(lli  +  iV  V6303) 
H-co^^(lli-J.V6303), 


and 


in  which 


(D 


=  1 


or  -i(l±V3). 


4.    Find  the  cube  roots  of 
-10  +  9V3. 


Assume     J  (y  +  aV^)  -=  (-  10  +  9  2V3)* 
in  which 


=  a>^(-10  +  9«V3), 


(1) 


an( 


a 


(I) 


-1. 

=  1  or  -  Kl  +  V3)  or  -}(1  -  ^3), 


and  therefore 


0) 


1  or  -  1(1  -  iV3)  or  -  i(l  +  «V3)- 


i(2/  -  aV2)  =  <o'  -^(-  10  -  9iV3). 


(2) 


302 


CUBIC   AND   QUARTIC   EQUATIONS. 


!  '■' 


(1)1(2),   y  =  a)^(-10+9/V3)+'^'^(-10-9;v3).       (3) 


(3)' 


y'---20-f  21y. 
(2/-l)(y~4)(7/  +  5)-0. 


yi 


y-i 


2/= 


=  -5. 


(i)x(2),  \(,f-z)  =  n. 

.•.z  =  2/»-28. 


V 


27, 


12, 


.-.  2i  =  — Z/,       Z2  =  — l^i       Z3  =  — O. 

(1) -(2),   aVz  =  o)^(-10+9zV3)-a)*^^(-10-9iV3), 
(7)',  az^z  =  18iV3  -  21aV2. 

Substitute  for  2;  its  values  given  in  (6). 

.•.a-^0i  =  — 3iV3,  aV02  =  2^v'3,  aV-3  =  ?V3- 
From  (4)  and  (8), 

(_  10  +  92- V3)i  =  i(l  -  3 1 V3), 
or  2  +  iV3, 

or  i(-5  +  ^V3). 

5.    Fijid  the  cube  roots  of  4  +  43iV5. 

Assume     |(2/+aVz)  =  w^(4+43zV5). 
.-.  |(y-aVz)  =  <«>^A/(4-43iV5). 
(l)4-(2),  y  =  0,^(4  +  43iV5)  +  <^"'-</(4  -  43zV5). 
(3)»,  2/^  =  8+63y. 

.•.(y--8)(2/'  +  8y  +  i)  =  o. 
.•.yi  =  8, 

ya  =  -  4  -  V15. 
(l)x(2),  \{f-z)^2\. 

.•.2  =  y2_34 


(4) 

(5) 
(6) 
(7) 


(8) 


(1) 

(2) 
(3) 

(4) 
(5) 

(6) 


CUBIC    AND    QUARTIC    EQUATIONS. 


303 


'iV3).       (3) 


(4) 

(5) 

(6) 

3-9iV3),  0) 


yz,=WS.  (8) 


43  iV^)- 


(1) 

(2) 
(3) 

(4) 

(5) 


(6) 


.•.Zi-=-20, 

Za-- 53 -8^15,  (7j 

23=:._53  4-8Vl5. 
(l)-(2),   aV2  =<u■^(4  +  43^V5)-a>V(4-43^VS).  (8) 

(87,  a2V2=86zV5-G3aV2- 

bo  +  ?: 
.-.  ay'2;l  =  2^V5, 

/    ^     86zV5     _43^V5(5  +  4V15) 
""■v^:^     10-8V15  25-240 

=  -^(V5  +  4V3), 
aV23=-^(V5-4V3)• 
Substituting  in  (1)  these  values  of  3/  and  a^2, 
i(2/i-t-aV2i)  =  4  +  ^V5, 
|(y,  +  aV^.)  -  i(-  4  +V15  -  V^  -  4^ V3) 

--i(l+V3)(4  +  w5), 
i(2/3  +  aV^H)  =  -  4(1  -  W^)  (4  +  V^)- 

Ex.  68. 

Solve  the  following  equations : 

1.  x^  —  Sx''  +  9x-6^0.  7.  x^-\-Sx^+x+l  =  0. 

2.  x^-Sx''-\-9x-9  =  0.  8.  3a;'  +  27a:'-9a;+41=-0. 

3.  2a;'-6a;'^+18a:+17  =  0.      9.  3a:'  +  27a;^-9a;+4  =  0. 

4.  a;3-3a;'»-15a;-13  =  0.      10.  a;^-18a:-33^  0. 


5.    a;'-3a;''- 15a;- 25  =  0.       11. 


.r 


'+9:i;'+9a;-fl5  =  0.        12. 


X 


x' 


^x- 


12  =  0. 


6a;H10a;-l  =  0. 


13.  x'^-{-y^=^6,  and  x^  +  y^  =  l. 

14.  o:^  +  y'^  =  6  wn,  and  a:'  +  y'  =  '^^^ (27n  — -  n). 


304 


CUBIC   AND    QUARTIC    EQUATIONS. 


15.  r'-Sx'-j-9x  +  (k-S)(l-{-k-')-^0, 

16.  x'  -Sx^-Wx  +  iki-  216)  (1  +  k-')  -  200  =  0. 

17.  x'  +  9a:^  +  9a:  -  3  (^  +  24)  (1  +  k~')  +  IG  --  0. 

18.  .t'- 2a:- 5  =  0. 

Find  the  cube  roota  of: 

19.  -8  +  6V3.  20.   -55-126iV3- 

21.    5  — 72V5. 

Solve  ax^  -{■  3  bx^  ~f  3  ^a:  +  ^  =  0,  given : 
22.    bd=c\        23.    2a''bd=  a'c^  +  b\ 

24.  a'bcd  =-- a'c^b^ -{- ac)  ~  b\ 

25.  kG-\-  H^  =  ¥,  h  being  arbitrary. 

26.  Show  that  IT—O  is  the  condition  that  ax^ -}- 2bx -\- c 

shall  have  a  square  factor ;  and  that  A  =  0  is  the 
condition  that  ax^  -{-Sbx'^  -^-i^cx  -i-  d  shall  have  a 
square  factor. 

27.  Show  how  to  solve  the  cubic  by  assuming 

ax'  +  Sbx^  +  Scx-^-d 

=  m(ax  +  b-i-i,y-}-(l—m)(ax  +  b  +  Q\ 

and  determining  m,  ti,  and  ^2- 

28.  If  ti  =  (xi  +  ciX2  +  oy^XsY  ^^^  ^2  ~  (^1  +  *^'^2  +  t^^'a)'* 

where  o>^  +  w  + 1  =  0,  find  ^i  +  ii  and  ii  t^,  and  apply 
the  result  to  solve  the  cubic. 

29.  If  Xi,  X2,  and  X3  be  the  roots  of  a  cubic,  express 

(a:i  —  a^a)' (a^a  —  a^a)^  (a:3  —  a^i)'  in  terms  of  the  coeffi- 
cients. 

30.  Prove  that  if  all  three  roots  of  a  cubic  are  real  and 

unequal,  A  will  be  negative ;  but,  that  if  two  of  the 
roots  are  complex,  A  will  be  positive. 


CUBIC    AND    QUAIITIC    EQUATIONS. 


305 


0  =  0. 
=  0. 


.26iV3- 


A  =  0  is  the 
shall  have  a 


I^i,  and  apply 


[press 

)f  the  coeffi- 


tre  real  and 
lif  two  of  the 


§  53.   The  Quartic. 

Let  the  general  quartic  equation  be 

ax*  +  4  bx'  -f  6  ex'  -\-4:dx  +  e  =  0.  (21) 

Assume 

a  (ax*  -f-  4  ir*  +  6  cx'^  -f  4  dx  +  e) 

=  (ax^  +  2bx  +  c  +  2ty~  (2^rx  +  s)\  (22) 

Expand  and  equate  coefficients  of  like  powers  of  x. 

.\r--=at-(ac-^b-),  (23) 

s^r  ^.2bt-  {ad-  be),  (24) 

s^=.U^-\.^ct-{ae-c').  (25) 

:.[at-  (ac  -  b'')]  [^f  ~\-^ct-  (ae  -  c')] 

=  [2bt-(ad-bc)]\ 
,-Ai''~(ae-4:bd+Sc'')t 
+  (ace  +2bcd-  ad'  -  eb'  -  c')  =  0,  (26) 

which  may  be  written 

^e-It+J=0,  (27) 

in  which  I=ae-4:bd+S e\  (28) 

J=  aee -\-2bcd-  ad""  -  cb'  -  c\  (29) 

Selecting  any  one  of  the  three  vahies  of  t  determined  by 
the  cubic  (27),  the  corresponding  value  of  r  may  be  found 
by  substitution  for  t  in  (23),  and  then  that  of  s  by  substitu- 
tion in  (24),  or  if  r  =  0,  in  (25)  ;  and  the  quartic  in  (22) 
may  then  be  resolved  into  the  quadratic  factors, 

ax'-\-2ib--y/r)x-\-e-\-2t-s, 
and  ax'  +  2{b-{-'^Jr)x^e  +  2t-\-s.  (30) 

Each  of  these  factors  equated  to  zero  will  give  a  pair  of 
the  roots  of  the  quartic  equation  (21),  which  will  thus  be 
completely  resolved.  (31) 


30G 


CUBIC  AND  QUARTKJ  EQUATIONS. 


The  equation  (27)  is  called  the  Reducing  Cubic  of  the 
Quartic  (21). 

Examples. 

1.    Solves:* -12  a;' +  47  a;'- 66:^  +  27  =  0.     (See  Ex.  41.) 
Let  a;*  -12r'  +  47a;'-66a:  +  27 


—  si 


6  a;  +  ^ -f  2  A'- (2  V^^  +  s)' 
■*  -  12ar'  -f-  /'se  +  ^+  4  iJ  -~\i\x^ 


(x^ 


2209 


-(94  +  24/;  +  4sVO^+^  +  -V^  +  4i5'-5'. 

DO 

Equating  coefficients  of  like  powers  of  a;, 

47  =  36  +  ^  +  4r-4r, 
3 

66  =  94  +  '^42!  +  4.sV^i 

36 
.•.6r  =  6i;  +  7, 
sV^'  =  -(6M-7), 
36  s^  =  144/^  +  1128/  -f  1237. 
(6Jf+7)(144^;''+1128^:  +  1237)  =  216(6/  +  7)^ 
6^+7  =  0, 
144  ?!^  +  1128  i;+ 1237 -216  (6^5 +7). 


or 


and 


144/^-168^-275-0. 
(12/ -  25) (12/+ 11)  =  0. 


t  _25 

Lit  > 

'      12 
13 


si'  =  ^»         s.,  =-3V13, 


11 

~12' 


^3  =  -. 


=  -3. 


0 


OJ 


CUBIC    AND    QUAUTIC    EQUATIONS. 


3U7 


X* -  I2x'  +  ^7 x' - Q(jx -\-21 


=    X 


{ 


,<> 


=  (x'-6xi-l2y      13{x-3y 

=:(x'~6x+ Gy  -(x-  sy  =--  0. 

The  last  gives 

x''  —  7x-\-0  =  0    or  a:'   -  5a; -f  3  =  0. 
.-.  a?  --  i(7  rir  V13)  or  a;  =  J(5  ±  y/lS). 

2.    Solve  9.^*  -  54a;'  +  60a;'  -  12x  +  16  =  0. 


27 
Here        a  =  9,  i  =  — — ,  c  =  10,  d  = 


18,  c=16. 


or 
If 


or 


or 


.'.  r  -  9  j!  -  (90  -  182i)  =  21  (4  iJ  +  41), 

8y/r  --=-271  -  (-  162  + 135)  -  -27{t~  1), 

s'=:4^^  +  40<;-(144  -100) 

=  4(«;'  +  10j5-11)  =  4(/  fll)(<-l). 
.-.  9(4^ +  41)(^  +  ll)(i;-l)  =  729(^5-1). 
.-.<!-  1  =  0, 

(4^  +  41)(^  +  11)==81(j5-1). 

j;=zl,  .•.r  =  101i  ands  =  0. 
.-.  (9a;'^  -  27a;  +  12)=^  -  405a;'  =  0. 
.-.  3a;'-(9  +  3V5)a;4-4-0, 

3a;'-(9-3V5)a;  +  4-0. 
.-.  a;  =  i  [9  +  3  V5  ±  V(78  +  54  V5)], 

x  =  \l^-  3 V5 i  VC^S -  54 V5)]. 

Ex.  69. 

Solve  the  following  equations  : 

1.  a;*  -  6a;' -  2a;' +  36a; -24  =  0. 

2.  a;*  -  2a;' -  25a;' +  18a; +  24  =  0. 


308 


CUBIC   AND   QrARTIC   KQt^ATIONS. 


3.    2x*~6x'-nx^-\-b3x~28^0.     6. 


a;' 


12a;- 5  =--0. 


4.    a;*-f-14a;'  f48a:  +  49=:0. 


6.  a;*-12a:-17=-0. 


7.  a;* -Br'- 12a:' -f- 84a;- 03  =  0. 

8.  a;*  +  2ar'-37a;'   -38.r+l  =  0. 


9.    121a;*  +  108ar' 

X 


100a;'-3Ga;4-4=-0. 


10.    a:'  +  y  =  li,  x-i~i/^2l 

§  54.  The  cubic  in  (27)  will,  in  general,  give  three  values 
of  t.  Let  them  be  denoted  by  <,,  /g,  4-  Also  let  the  corre- 
sponding values  of  r  and  cf  s  be  denoted  by  r,,  r.^,  r^,  and 
•"?i.  ^2)  s-d,  respectively.  Let  Xi,  x.^,  x^,  and  x^  denote  the  roots 
of  the  quartic.     Then,  by  (31), 

ax""  +  2(6  -  Vn)a;  +  c  +  2ti-8i  =  0  (32) 

will  furnish  a  pair  of  the  roots  of  the  quartic  (21),  say  a;,,a;2, 

^^^^  aa;'  +  2(i  +  -y/r^x  +  c  +  2^^  +  Si  =  0 

will  furnish  the  complementary  pair,  x-i,  Xi. 

So  also  ax^-{-2(b~  v'r,)  a;  +  c  +  2  A,  -  Sa  =  0  (33) 

will  furnish  a  pair  of  roots  different  from  either  of  the  above 
pairs,  say  Xi,  Xj,  and 

ax'  +  2{b-{-^r,)x  +  ci-2t,-{-s,=^0 
will  furnish  the  complementary  pair  a;^,  x^. 

Finally, 

ax^  -\-2{h-  ^r,) a.-  -f  c>  +  2 4  -  S3  =  0 

will  +"  .^'nish  a  pair  of  roots  different  from  any  of  the  pre- 
ceding pairs.  These  must  therefore  be  either  a^i,  x^,  or  else 
Xi,  X3.     Then 

ax'  +  2(b-}-^r,)x  +  c  +  2i,-\-s,  =  0 

will  furnish  the  complementary  pair ;  that  is,  either  x^,  x^, 
I  the  case  may  be. 


'ij  ^ii 


CUBIC    AND    QUARTin    KQUATIONS. 


309 


5  a;- 5-0. 


Ja;-n-0. 


4-0. 

three  values 
let  the  corre- 
ri,  r.i,  n,  and 
lote  the  roots 

21),sayx.,a;2, 
0 

(33) 
r  of  the  above 


of  the  pre- 
\Xi,  Xi,  or  else 


either  x^,  x-i, 


Let  y*=  1,  then  y  may  l)o/<o  (ietcrminod  that 

ax'-\-2(h~-y^r,)x-{-c  +  2h-ys,^0       (34) 

will  furnish  the  pair  of  roots  x^,  Xt,  and  then 

ax'  -I-  2  (/>  -f-  y  V''«)a:  +  (7  +  2/3  -f  ysj  =  0 

will  furnish  the  complementary  pair  Xj,  x^. 

By  (32\  x,-\-  x,--^  --{b  -  ^r^). 

By  (33),  a:,-f:r3---(ft- V^a)- 

By  (34),  x,  +  x,^-^(b      y  Vr,).  (35) 

Co 

45 

By  (21),  a^i  +  a:,  +  0:3  -f-  a:*  = 

a 

ax^-\-b=  V^i  —  V^-,  -  y  y/r^ ; 

aa;3  +  6  =  — v'^-,  + V^a -^yy'rsi  (36) 

axt  [-b  =  --  V^'i  -  ^?-a  -f  y  ^r3. 

Also,  from  the  first  throe  equations  of  (35), 

7  Vn  V^-^V^*3=^^  [^>'+^«(:i^i+^2)F>>+k<.r,+^3)]rH  J«(a:,+^-4)] 
=  5'  +  Ja5'^(3a:i  +  a:2  +  a73  +  a;4)  +  ^a'^Z»[(a:i+a:,)(a:i  +  a;3) 

+  (Xi+X3)(Xi~{~X,)'}-(Xi-{-Xt)(Xi-i-X2)] 

+  i  a'  (a:i +3:2)  (xi  +  0:3)  (a^i  +  x^) 

=^P~{-^ab'^(Sxi-}-x.i-}-X3~]-Xt) 

+  }  a^i  [3a:i'^-f-  2xi  (x\-{-X3'i-Xi)-\-x\X3-{-X2Xi+X3X\] 

+  i  a'  [a:i^ + xi'  (x., + 0:3  +Xi)-\-Xi(x.,x^-}--j:.iXi-\-X:iXiy\-x-,x.,x^  ] 

=^b'+fiab'(2x,-—)  +  Wb(2x,'--x,+~ 
\  a  J  \  a  a 

+  iia'(-~x,'~^==~-P+^abc~^a'd      • 

^-^a''d-Sabc  +  2b'). 


310 


CUBIC    AND    QUARTIC    EQUATIONS. 


|i  m 


(37) 


That  is,    y  -y/vi  V'2  V^'s  -^  —  hG. 
[See  (5),  page  297]. 

Therefore,  y  =  -f  1  or  —  1 
according  as  G  is  negative  07^  positive. 

Hence,  by  (4),  (5),  (21),  (23),'  (27),  (28),  (29),  (36),  and 


(38) 


(38),  if 
then  shall 


ax*  +  4  bx^  I  6  r.r'  i-  4  dx  |-  c  -^-  0, 


^3  =^  -  (-  b  ~  yji\  +  ^i\  -  -  y  V^O. 

3^4  =  -(-  ^  -  V'*i  -  V^'2  +  y  V^'3), 


(39) 


in  which      ri^=ati  —  H, 

r^  =  at,-  II,  (40) 

rg  =  at^  —  H, 
ti,  ti,  4  being  the  roots  of  the  equation, 

4.t'-It-\-J=^0,  (41) 

H—  ac  —  b^, 

I  ^ae-4:bd+^c\ 

J  =  ace  -f-  2  ic'(/  -  a(^^  —  eb^  —  c', 

6^=:a^f?-3a^»c?  +  2/>^ 
and  y  =  -f  1  or  —  1  according  as  G  is  negative  or  positive. 

§  55.    The  roots  given  in  (39)  may  also  be  expressed  in 
terms  of  any  one  of  the  three  values  of  t,  as  follows : 

By  (40),  n  +  r,  +  v.,  -  a (t,  4- 1,  +  t,)-~?>B'. 


CUBIC   AND   QUAUTIO   KQUATIONS. 


311 


(37) 


(38) 


29),  (36),  and 


)- 
). 
.). 
a). 


(39) 


(40) 


(41) 


Ive  or  positive. 

)e  expressed  in 
follows : 


But  (41),  t^  -f  t^  +  i^a  =  0. 

Now,  V^'a  +  7  V^'3  =  V(^'i  +  ^'3  +  2y  V^*2  V^a) 


V 


(— 


37/ 


JL 


See  (37). 


In  like  manner  it  may  be  shown  that 

V.-yV3  =  ^(-'-.-37/+^_)- 

Replacing  rj  by  ati  ■—  If,  see  (40),  and  solving  (41),  the 
result  becomes 


If 

then 


Xy 


X. 


aa;*  4-  4  5a;^  +  6  ca;'  +  4  c?a;  +  e  =  0 , 

l{-6+V(«^-//)+4-a^-2^--^ 
h^y/{at'Il 


a  I 


■n[- 


-a^-2  TT- 


■^{cii-H)_ 


a 


in  which      t  =  -tA\  J-\- 


~at-2  II- 


V(27/= 


^{at-II) 
G 


3V3 


3V^ 


V(277^-P) 


=-14- 

^^  ac  -  h\ 

I  -=ac-46c^+3c^ 

J"  =  ace  +  2  See?  —  ad^  —  ei''  —  <r* 

(^  =a'J_3aic4-2il 


^{at-II) 


} 
) 


:(i^') 


(43) 


312 


CUBIC    AND    (iUARTlG    EQUATIONS. 


Ex.  70. 

1.  Reduce  the  quartic  ax*  +  4  hx^  -\-  Q  cx"^  -{- 4:  dx -{-  c  =  0 

to  the  form  ?/*  +  6^2/"^  +  4  Gy-\-aU-  2>IP  =  0. 

2.  Show  that  the  two  quartics  x*  -\-  G  Hx^  zh4Gx-\-  K=  0 

have  the  same  reducing  cubic. 

Solve  the  quartics  : 

3.  :r*-24a;2  =±=3207-132  =  0. 

4.  x*-%x^±20%x~  2^21^0. 

5.  :r*-6:i;'±  16^  —  33  =  0. 

6.  ^r^-eoj'^rfc  16a; +  39-0. 

7.  x*-(Sx''±4Sx  —  \l1    -0. 

8.  .r*  +  6a;'^  — 60a; +  36  =  0. 

Show  that,  if  Xi,  x^j  a^g,  a;^  be  the  roots  of  a  quartic, 

9.  48  ^=  —  :S  (a;i  -  a;2)'. 

10.  24a''I=%{x^  —  x;)\x^~x,)\ 

11.  32  (r  =  ±  (a^i+a^a— a;.,— a:4)(a;i+a:3— a:*— a:2)(ri+a:4— a^a— a^a). 

12.  432  a'V=  [(a^i  —  a^j)  (x^  —  0:4)  —  (a^i  —  x^  {x^,  ~  x^'] 

X  [{xx  —  x^)  (a:4  —  X2)  —  (a:i  -  a;4)  {x.^  —  ajg)] 
X  [{xi  —  x^  (a?2  —  x^)  —  (a:i  —  a;.,)  {x^  —  x,)\ 

13.  4:JP-a'IH'+a'J+G'  =  0. 

14.  Prove  that,  if  27J^  =  /^  the  quaitic  has  a  pair  of 

equal  roots. 

15.  Prove  that,  if  /=  J=~0,  the  quartic  has  three  equal 

roots. 

16.  Prove  that,  if  o?  1=12 IP,  and  a^J=8IP,  the  quartic 

has  two  distinct  pairs  of  equal  roots. 


CUBIC   AND   QUARTIC   EQUATIONS. 


313 


4  r^o;  +  c  =  0 
3iP  =  0. 


quartic, 


':>^{x.,-x,)\ 

I  has  a  pair  of 
las  three  equal 
IP,  the  quartic 


18 


19. 


Solve  the  quartic  x*  +  G  Ihr  -f  4  Gx  +  d'l-  3  IP  --  0  : 
17.    By  reducing  it  to  the  form 

{x"  +  2^yx  +  z,)  {x'  ~  2^yx  +  2.)  -  0. 

By  reducing  it  to  the  form 
2/2  (^'  +  yxx  +  zf  -  ?/,  {x'  +  y..^-)''  =  0. 

By  reducing  it  to  the  form 
{x'  +  yj  -  (z,x  4-  z,y  =  0. 

20.  By  assuming  the  roots  to  he  of  the  form 

in  which  a^  -  1,  P"  =  1,7'  =  1. 

21.  By  assuming  the  roots  to  he  of  the  form 

in  which  i'"*  +  1  =  0,  and  n  is  integrah 

22.  Apply  the  method  of  Exam.  20  (Euler's  Method)  to 

solve  the  quartics 
x'—Qlx^-±^y/{P-Ym^-]-7v'-^lmn)x--^{^mn-l')-==^0. 

23.  Reduce  the  quartic  to  the  form  3/^+6  Cy"^  +7^  —  0,  hy 

assuming  x  =  ~^—^,  and  suitably  determining  z^ 
and  Z'l. 

24.  Reduce  the  quartic  to  the  form 

2/*  +  4j5/  +  GQ/^  +  4i?y  +  l  =  0; 

by  assuming  x=^Zx-\-z.iy,  and  suitably  determining 

Zx  and  z^. 

Make  the  same  reduction  as  in  the  last  question,  by 
assuming  a:  =  2;i  +  2.2y~\  and  suitably  determining 
Zx  and  Z2. 

Eliminate  x  between  a;'  +  6^a;'  +  4(?a;-fa'^/-3/r  -0 
and  a;''  +  23/:r-l-2;  =  0,  and  so  determine  y  that  the 
resulting  equation  may  reduce  to  the  form 
z'-\-Q>Cz^-\-E^^Q. 


25. 


26. 


314 


CUBIC   AND    QUARTIC   EQUATIONS. 


If  Xi,  X2,  ^^3,  x^  denote  the  roots  of  the  quartiu 
x'  +  6IIx'  +  4  Gx  +  a'l-SH'  --=  0, 
form  the  cubic  whose  roots  are  : 


mo.    X\X2  ~r*  x^x^j  X\X^  ~\~  x^x^y  x^x^  ~j~  X2X^ 


29. 


'l^'Z 


3'*' 4 


u7i  ►t7"»  CL  aX 


4'<'2 


^|u<7^    "~~    JOnJO^ 


-\-Xi 


X.J  ---  x\ 


Xi  ~\~  X2        X^        X^      Xi  ~j~  X^    ~  X^        X2      X\ 

«5u.    {x^X'i      '^3*^4/ ( "^1  ~t~ ''^'i      ^3  —  '^iji  etc. 

31.  (^Xi     X2)  {X3     Xi)  ,  {^Xi     X3)  (3^4     X2)  ,  (^1    x^)  (^2    x,i)  . 

32.  i^Xi      X2)  {X2      x^)  (^3      x^)  {^x^      Xi), 

{^Xi        X'^)  (cl'3        XaJ  {^Xi        X2)  {X2        Xi), 
(.Ti        X'ij  (^'4        X2)  (Xj         Xi)  (.^3        ^^1  j. 

33.  Show  how  to  solve  the  quartic,  knowing  the  roots  of 

any  of  the  above  cubics. 

34.  Reduce  ^ach  of  the  cubics  in  Exams.  27  to  32  to  the 

standard  form     j/^  +  Ci/  -\~  D  =  0. 

Form  the  equation  whose  roots  are  : 

35.  The  squares,  36.    The  cubes, 

of  the  roots  of  ax^  +  3  bx'  f  3  ex  +  d  ^  0. 

Form  the  equation  whose  roots  are  : 
37.    The  squares,  38.    The  cubes, 

of  the  roots  of  ax^  +  ibx^  +  G cx"^  -\-4:dx-{-  e  =  0. 

39.  Form  the  equation  whose  roots  are  the  squares  of  the 

differences  of  the  roots  of  a  cubic. 

40.  Form  the  equation  whose  roots  are  the  squares  of  the 

differences  of  the  roots  of  a  quartic. 


X^ A*2        373^  . 


1^4  XiiX^ 


■  X^        X'2        X^ 


-X^j  \X2     Xa)  . 


the  roots  of 


7  to  32  to  the 


s, 
^0. 


quares  of  the 
quares  of  the 


CHAPTER    X. 
Determinants. 


I.   Definitions  and  Notation. 


§  56.    The  symbol 


«2     ^2 


denotes  the  expression     aj)~i  —  ajbx, 
which  is  called  a  Determinant  of  the  Second  Order. 
The  symbol 


denotes  the  expression 


rti 


«1 

b,   Ci 

«2    ^2    C'i 

ession 

«3    ^3    Ci 

h    <?2 

—  «2 

h    ^1 

-  ci-.i  !  bi  Ci 

^3   ^3 

h    C3 

\  b,  c, 

which  is  called  a  Determinant  of  the  Third  Order. 

The  symbol 


(Xi    b^    Ci 

a^   ^2   ^2 
«3    b-i   c^ 


h 

k, 

h 


Ctn    "'n     ^»t  "^'« 


denotes  the  expression 


(U 


b-i  c,     hi 
b-i  ^3 


h     n       I' 


-  a~y 


bi  Cx     hx 

^3    ^3      "-3 


b,,    Cn 


+ h(-ir\, 


bi     c, 
bo     c 


2  -^2 


^H  -1  ^«-l  ^n-1 


which  is  called  a  Determinant  of  the  nth  Order. 


316 


DETERMINANTS. 


Examples. 


1. 


2. 


a-\"b     a  —  h 
a  —  h     (I  -f  h 

1  X    y    \  = 
1  x'   y' 
1  x"  y" 


=  (a-^hy  -  (a~  by  =  ^ab. 


x'    y' 

x"  y' 


X    y 
x"  y" 


-f 


X    y 
x'  y' 


_  ^fyfi  __  rfjf'yi ._. xy''-^- x"y  -f-  xy^   - x^y. 


3. 


12  3 

=^  1 

3  4 

-2 

2  3 

o 

2  3 

2  3  4 

4  5 

4  5 

3  4 

3  4  5 

=  1(15-1^    -2(10-12)  +  3(8-9) 
=  -1+4-3-0. 


4. 


0  rt  /v  c 

a  0  z  y 

b  z  0  X 

c  y  X  0 


--  —  a 

a  b  o 

+  h 

a  b  c 

—  c 

a  b  c 

z  Q  X 

^  z  y 

^  z  y 

y  X  0 

y  X  0 

z   0  X 

=  +  d^x^   —  aZ>  j:?/ 

-\-  b^y'^   —  abxy    —  bcyz 
~\-  chf —  bcyz 


cazx 


cazx 


d^x^  +  bhf  -\'  &'^  -  -  2  abxy  —  2  bcyz  —  2  cazx 


Ex.  71. 

Expand  the  following,  i.e.,  write  them  in  ordinary  alge- 
braic notation : 


1. 


a  y 
b   X 


2.    \a  b 
\y  X 


3. 


y  a 
X  b 


4. 


b   « 
X  y 


5. 


ma  y 
mib  X 


ma  my 
b       X 


7. 


a  my 
h  inx 


8.    I   a      y 
i  ')nb  -mx 


9. 


a  +  my  y 
b  -f-  mx  X 


10. 


a  -\-  mb  y  -\-tnx 
h  X 


11. 


a,  —b 


a 


[ah.    I 


/'    x^y- 


8-9) 


c  \a  h  c 

0  z  y 
z    ^  X 


cazx 

cazx 
\cazx 


)rdinary  alge- 


4. 


h  a 
X  y 


18.    I  a      y 
\  mh  mx 


.1. 


a, 
a'. 


-h 


DETERMINANTS. 


317 


12. 


15. 


x"^  +  a?       ah 


ah 


X' 


-H  h'^ 


<f\ 

^2 

«3 

h. 

b. 

^3 

Cx 

C'l 

c^^ 

13. 


16. 


ah        c^ 
-  cC'  —  he 


(h  h-A  c^ 
a-i  h,  c, 
a,  hi  Ci 


14. 


a  -  b,  —  2a 
-2b,b-a 


17. 


18. 


1 

1 

1 

X 

y 

z 

a 

b 

c 

19. 


1 

1 

1 

a 

i 

c 

.  «' 

h' 

c'^ 

20. 


X  y 

IH 

Z     X 

y 

y  - 

X 

a  X 

y 

X  a 

y  z 

a 

21. 


23. 


29. 


• 

a-^b 
a 
h 

c 

b-\-i 
h 

c 
?      a 
c- 

-a 

22. 

1       1           1 
1   1  +  x      1 
1      1      l+y 

Tna^  vihi 
as      ^3 

7)1  Ci 
C-A 

24 

(ii  4  7r 

^'3 

<^3  hi-\-vib.i  Cj-f-w^a 
h.,               c. 

25. 

^2 
«3 

a. 

hi  c 
hi  c 
hi  c 
hi  c 

I     C?2 

s  d; 
K  d^ 

26. 

• 

ai  aa  ^3  a^ 
hi  h,  h,  bt 

Ci     Ci     ^3      t?4 

di  d-i  di  di 

27. 

1 
rt 
a^ 
a' 

1    1 

h     0 

y  c 
h'  c 

1 

d 

28.    ' 

X      0      0    ^3 

—  yx       0     ^2 

0     —3/     X     rti 

0       0    —  3/  tto 

1 
1 
1 
1 

1 

1  + 
1 
1 

X 

1 
1 

1+? 
1 

1 
1 

/    1 

14 

-  z 

30. 

1- 
1 
1 
1 

-  a      1         ' 

1  +  ^    : 
1    1- 
1      : 

L 

L 

-c 
L      1 

1 
1 
J 

-\-d 

§  57.  The  quantities  which  in  the  determinant  notation 
stand  unconnected,  and  which  are  taken  as  factors  to  form 
the  terms  of  the  expanded   determinant,   are   called  the 


318 


DETERMINANTS. 


Fjlcmcnts  of   the    dctorminant ;    e.g.,   the    t'lemeiits  of  the 
(iGtermlnant 

a  +  h      a  h 

c      c-\-  d     d 
e         f      e-\-f 

are  the  nine  quantities  a  -\-  h,  a,  h,  c,  o  -f-  d,  d,  c,f,  c  +/. 

The  elements  standing  in  any  horizontal  line  constitute 
a  How,  and  those  standing  in  any  vertical  line  constitute  a 
Column.  The  rows  are  numbered  first,  second,  third,  etc., 
beginning  at  the  top,  and  the  columns  are  similarly  num- 
bered, beginning  on  the  left.  The  elements  of  a  row  are 
numbered  first,  second,  third,  etc.,  beginning  on  the  left, 
and  those  of  a  column  are  similarly  numbered,  beginning 
at  the  top.  Hence  the  ?/it]i  element  of  the  ??th  column, 
called  the  (?//,  n)th  element,  is  the  wth  element  of  the  mth 
row;  it  is  often  denoted  by  am,n,  the  first  suffix  denoting 
the  row,  and  the  second  the  column  to  which  the  element 
belongs.     Thus,  in  the  above  determinant, 


Cli 


•i,3 


d,    «3,2=y,    a-,^2  =  o-yd. 


A  determinant  is  said  to  have  two  diagonals,  called,  re- 
spectively, 2^'>'i')^(^ipctl  and  secondary.  The  elements  stand- 
ing in  a  line  from  the  upper  left-hand  corner  to  the  lower 
right-hand  corner  constitute  the  principal  diagonal ;  those 
standing  in  a  line  from  the  upper  right-hand  corner  to  the 
lower  left-hand  corner  constitute  the  secondary  diagonal. 
Thus,  in  the  determinant  given  above,  the  principal  diag- 
onal elements  are  a-\-b,  c-\-d,  6'+/,  and  those  of  the 
secondary  diagonal  are  h,  c  -\-  d,  e. 

The  produjct  of  the  elements  standing  in  the  principal 
diagonal  is  called  the  principal  or  leading  term  of  the 
determinant. 

Where  there  is  no  danger  of  ambiguity,  a  determinant  is 


eiits  of  tlie 


le  constitute 

constitute  a 
l^  third,  etc., 
lilarly  num- 
)f  a  row  are 

on  the  left, 
>d,  beginning 

7?.tli  column, 
it  of  the  wth 
,ffix  denoting 

the  element 


Is,  called,  re- 
ments  stand- 
to  the  lower 
gonal;  those 
orner  to  the 
ry  diagonal, 
incipal  diag- 
hose   of   the 

the  principal 
I  term  of   the 

jterminant  is 


DETERMINANTS. 


310 


often  denoted  by  writing  only  its  principal  diagonal  elo- 
nients  between  vertical  bars  or  within  parentheses.     Thus, 

I  ctx  b-i  6*3 1  denotes 


11^  bi  Ci 

a^  ^2  ^2 

as    ^3    <^3 

a     X    y 
a'    .X-'    y' 
r/'   .t"  y/" 

^5  (Jr.  h 

C'l  g%  h 

<^o   (Jo   "-0 

a  x'  y"  I  denotes 


and  1 6*5  (jf2  ko  \  denotes 


If,  in  any  determinant  of  order  )(,  \]\q  pth  row  and  the 
^th  column  be  erased,  and  all  the  rows  above  the  />th  and 
all  the  columns  to  the  left  of  the  qth  be  transferred  in  order 
over  the  others,  tie  resulting  determinant  multiplied  by 
(—  !)(''-  '>(p+«>  is  called  the  complement  of  the  (p,  q)th.  ele- 
ment. 

If  71  be  odd,  (-  l)(«-i)(p+9)  =  1  for  all  values  oip-\~q. 

If  the  elements  of  any  determinant  are  each  denoted  by 
a  small  letter,  and  are  all  different,  the  complement  of  any 
element  may  be  denoted  by  the  corresponding  capital  letter 
affected  with  the  suffix  or  suffixes  of  the  element. 

Thus,  in 


«1 

il      Cy 

«2 

h     C'l 

"3 

h  r'3 

Ax  denotes  the  complement  of  «!,  which  is 


A>i  denotes  the  complement  of  a^,  which  is    h^  c^ 

Ih  G\ 

B^  denotes  the  complement  of  Z>3,  which  is  I  c^  cii 

\c.,  a., 

In  these  n  —  l  =  2,  and  .-.  (--  l)("-^)(^+<')  -..  1. 


320 


DETERMINANTS. 


In 


A  =  - 


h. 

Cs 

d. 

K 

(\ 

d, 

K 

Ci 

d. 

a,  />,  (\  d^ 

Ui  />a  Ci  (l^ 

"■.i  f>\\  (^w  d:x 

(It  h^  c^  di 


IToro  ?2  —  1  =  3,   p- 


-1.    <I 
-1. 


—  9 


and  On  = 


d. 

«4     ^4 

d, 

<h   ^h 

d. 

Ch   h 

Here  n—\~  3,  j^  —  3,    q 
and  .-.  (-  l)("-»)0'+«)  ^  1. 


-3. 


Written  in  tlie  diagonal  notation,  the  last  example 
would  be  : 

In  I  ax  h-i  Ci  d^\     A2~    - 1  Ih  c^  d^  \    and   C^,  =  |  d^  cix  h^  | 

To  find  the  complement  of  a  product  of  two  or  more 
elements,  find  first  the  complement  of  one  of  the  elements ; 
then  in  this  complement  find  the  complement  of  a  second 
element  of  the  product ;  next,  in  this  second  complement 
find  the  complement  of  a  third  element  of  the  product ;  and 
proceed  thus  through  the  whole  product.  The  final  com- 
plement will  be  the  one  required. 

For  example,  to  find  the  complement  of  aa^s,  first  find 
A-i ;  then  in  A.^  find  the  complement  of  h^.  Similarly,  to 
find  the  complement  of  a^bid^,  first  find  A4 ;  then  in  A^  find 
the  complement  of  bi ;  then  in  this  complement  find  the 
complement  of  d^..  The  order  in  which  the  partial  comple- 
ments are  found  will  affect  the  form,  but  not  the  value  of 
the  result.  Thus  the  expansion  of  the  complement  of  a^bids 
will  be  the  same  whether  found  in  the  order  a^,  bi,  d^,  or  in 
the  order  a^,  d-i,  b^,  or  again,  in  the  order  (^3,  5i,  a^. 

The  7?ith  element  of  the  nth  column  is  called  the  con- 
jugate of  the  7it\i  element  of  the  mth  column,  and  vice  vei^sci; 
i.e.,  an,n  and  a„  „,  are  each  conjugate  with  respect  to  the 


/, 


■■J 


DETERMINANTS. 


lil^l 


=  1, 

<1  = 

=  2, 

-1. 

-3, 

7^ 

^3, 

:1. 

ast 

example 

=  1  iu  th  h 

two  or  more 
the  elements; 
,  of  a  second 

complement 
product;  and 

e  final  com- 

2^3,  first  find 
Similarly,  to 
len  in  A^  find 
lent  find  the 
^rtial  comple- 
the  value  of 
lent  of  aj)idi 
L,  K  t^3,  or  in 

[lied  the  con- 
oid vice  versd ; 
jspect  to  the 


other,  or  are  a  pair  of  conjugates.  Each  olomont  of  the 
principal  diagonal  is  its  own  conjugate  or  is  self-conjugate. 

A  symmetrical  determinant  is  one  in  which  each  clement 
is  equal  to  its  conjugate. 

A  skew  determinant  is  one  in  which  the  sum  of  cacih^^aiV 
of  (conjugates  is  zoro.  * 

A  skew  symmetrical  determinant  is  a  skew  determinant 
whose  principal  diagonal  elements  are  all  zeros. 

Ex.  72. 

Write  in  the  "  square  "  notation  : 
1.    I  «i  &3  Cr,\  2.    I  x.^  ?/i  z^\  3.    I  (Iq  (?3  ^2 1 

4.    I  Wq  Xy  7/2  Za  I     5.    I  a^  bi  c^  d^  \     6.    |  «,,  1  rtj.a  «3,3  «4.4 1 

In  I  «i  hi  C3  c?4  C5 1  find  the  complements  : 
7.    ^2.       8.  B,.       9.   ^5.       10.   A-       11.    Ci.       12.   E,. 

In  the  same  determinant  find  the  complements  of : 
13.  cuj)-}.    14.  axCi.    15.  a^Cy.    16.  aJj-iC^.    17.  /a/^^i.    18.  a-iCidi. 

Prove  that : 

19.  I  rti  ^2  C3 1  —  a^Ax  +  hiBx  -\-CxCx  =  axAx  +  a^A^  +  a^A^. 

20.  axA^  +  hxB.i  +  CxC^  =  axA-s  f  hxB^  +  CxC^ 

=  ttiBi  -j-  azBi  +  a^Bi  —  axCi-\-  ajC^  +  ^^3  —  0. 


II.   Transformatio  .•. 

§  58.  Theorem  I.  The  value  of  a  dcUrTninant  luill  7iot  be 
altered  if  the  columns  be  written  in  order  as  roios,  and  vice 
versd. 

Hence,  in  any  theorem  in  which  the  word  "row  "  occurs, 
the  word  "  column  "  may  be  substituted  therefor,  and  vice 


hI'n'KUMlNANTH. 


vrrsi) :    MUtl,    m    ;my    ll(i»(»r<Mii    \u    wlin'li    liolli    "  row   '    mikI 
*' oolmnn      oi'fur.  ihfMc  wiuds  iumv  Ito  mlt'icliiiiiiMMl  willioiil 

Tuii^UMM  11.  //  ti)}}/  /»/•!»  rrwr.s-  (or  fir<>  ('ohntnts)  of  o 
(fr/t'n»i))n})f  hr  mfrrr/hnKfrff,  thr  risit/fniif  if(  fi  rininiinf  \rill 
tiiiVf-r  iu>/)/  ni  sh/)i  h'om  thr  (wujnml  our. 

('«>I\.  1.  If  o  )'i^ii'  {(>)•  (1  <'o/)nnn)  hr  fmnsl'n'rd/  orrr  n 
i)//^,r  nui's  (<);•  ro!unnis),  (hr.  (/<f<'nnfn<inf  iri/l  he  nm/fi/i/nd 
/.// ^      IV. 

(\>K.  'J.  A  /ni))sfrr  of  p  <'t))isrrNfir('  roirs  (or  rohnnns) 
0V(')'  \\\  |>  of/)<r  (\)n,'<ri'ufivr  rows  (or  ro/inmis)  inuldplics 
fhr  Jrfn'wiwmf  hi/  (       IV"  '^'\ 

Con.  iV  1/  Ap, .,  (h'))o/c  thr  rrmpJrmrnt  of  thr  ([>.  "jV// 
rhnnrnt  of  tnn/  (hirrnn'nauf  A  of  orihr  n,  fi/;f/ A,,,,  (h'liotr 
the  (irtrrwinant  forwoJ  from  A  />v  striki)\(j  out  thr  \\th  ron^ 
and  the  i\th  co/umn,  then  will  A,,,,,      (     l)"'*"^A,,,.,. 


KxAMPliKS. 


1. 


./,  A,  (', 

<U  h,  (', 

ih  ^1  '\ 

, 

(T,  fTj  ff;, 

/),  A,  A., 

''i  <'-i  '',1 

=r+    A,  A,  A, 

(hi  Ox  'h 

('•2  <\  <\ 


A,   o,   r, 
A,  rr,  r, 

Am    <fa    f':i 


A,   A,  A, 
(I'l  ('-,  ff. 


Ci    r.i   rt 


2.    Transform  '  o^  A.,  r■^  |  so  that.  A.^cvifs  sliall  1)(^  llio  iirsl  row. 


a 


I     A,     f,      :^=      (h    A.,    r\, 


(7.2  A.^  e^ 
<h   A3  ('3 


ffi  A,  r, 
a,,  Aa  r., 


h    ^M    'C;. 

A,  ('1  a^ 
Aj  <'.,  (/,., 


or     I  (7i  Aj  ^3 1  =^  I  rTs  A,  e.^  \  -    I  A;,  f,  r).^  | 

3,    Tninsform  |  Oi  h.^  r,  |  so  that  A./r^ri  shall  bo  the  principal 
diagonal  elements  in  order. 

I  Ox  A2  Ci  i  =  i  cf'i  A3  (',  I  =  —  I  Aj  ^3  r,  I 


ttf-rrr-iiMrNANT; 


:\^\ 


row   "    Miwl 
rod  Nvitlm\»t. 


Dlithllll    U'lll 

or  colli  in  nf<) 
s)   iinil/i}ifi<'i< 

'  thr  d'.'p'^' 
///,■   )»///   row 


/»,  /',  /';, 
(»,  (''2  '':\ 
('l    (\    ''s 


(ho  first  row. 


the  principal 


II  A 


It 


K  '':.  'K  'ft  /il,   ill"'    ^ 


lUH 


1    J 


fl,  ft' 


II 


oi'o       n 


I.  ft 


lllKl 


:i,ft 


(     I) 


»1,  ft 


I  "i  f>>  '•«  '^ft/n 
1./.  "ft  ''' 


'n  '"i  '^j 


Ex    73. 


II 

h 

1' 

,1 

i' 

f 

!l 

h 

k 

TiauHruiiii 


iiild  tin  (*(|niv.'ili>iit,  (loloniiiniint  having: 

1.  //,  //,  /•  MM  ilM  lirHt  low. 

2.  A,  (\  h  iiH  ilH  lirMl,  row. 
W.  il,J\  r  )iH  IIh  lirHt,  i  >w. 

4.  /',  /.-,  r  MM  il.H  firHl,  r()W. 

5.  (',  <\  (J  MM  Hm  prin('i[>Ml  diM^oHM,!. 
^'  \uj^  ^*  iiM  itM  HCMUjiidury  diM|^oriMl, 

TrM!i»rorin  a  h    c   d 

k  I    n>,  11 
l>  y    ;•    .s' 

inlo  an  oqnivalnnt  doiorinifiMnt,  luivitif^ : 

7.  (1,  d^  r,  h  aH  its  firHt  row,  and  a,  k,  t\  p  an  iU  firHi 

colnnin. 

8.  .s',  //,,  A,  d  as  its  third  row,  and  ?//,  /,  w,  k  as  its  second 

column. 

9.  c,  in,  q,  (/as  its  princij)al  diagonal. 
10.    *',  7/i,/,  a  as  its  secondary  diagonal. 

Prove  that : 


11. 


a  b  0    c 

— . 

d  e   f   0 

0  0  r/    0 

h  k  I   m 

r/  0  0  0 1 

/  7//  A  /• 

0  c  a  h 

f  0  d  c 


ma 


824 


DETERMINANTS. 


12. 


0 


c    d 

g   h   k  I 

0  '»!  n  0 


b  0 


d  c  f  c 

a  O'O  b 

m  0  0  ?2 

h  (J  I   k 


13. 


d  a  c  b 
^000 
q  Q  f  c 
'k  0   h  0 


a 

0  0  0 

b 

e   0  0 

c 

f  hO 

d 

0  ^  ^ 

Transform 


a 

b 

c 

d 

b 

c 

d 

a 

c 

d 

a 

b 

d 

a 

b 

c 

so  as  to  have  the  principal  diagonal  composed  of : 

14.    The  four  a's.  15.    The  four  i's. 

16.    The  foiir  r's.  17.    The  four  (i's. 


=  -!^ 


Prove  that : 

18. 

«i 

h. 

— 

19. 

«i 

h. 

03 

20. 

«! 

b. 

C?i 

21. 

«1 

h 

C-i 

22. 

ttl 

h 

<?3 

«2! 

=  —  I  ^1   h.,  Us  I 
di\  —  \di  C.2  b^  tti  I 
c^  d^  C5  I  =:  I  ei  d^  C3  bi  fts  I 

c?4  ^5  /e  I  =  —  1/1  ^2  ^3  ^4  ^5  «6  I 
23.  If  two  determinants,  A  and  A',  of  the  nth.  degree  be 
such  that  the  first  row  of  the  one  is  the  same  as  the 
last  row  of  the  other,  the  second  row  of  the  one  the 
same  as  the  (71  —  l)th  row  of  the  other,  the  third 
row  of  the  one  the  same  as  the  (n  —  2)th  row  of  the 
other,  and  so  on,  then  will  A  =  (— l)i"<~-^^A'. 

Transform,  by  cyclic  transposition  of  the  rows  and  col- 
umns, the  determinant  |  «!  b^  c^  d^  <?5 1  into  an  equal  deter- 
minant having : 

24.    <?4  in  the  first  row  and  first  column. 


of 


1. 


detp:kminant.s. 


of 


iir  d'a. 


ith  degree  be 
|ie  same  as  the 
3f  the  one  the 
ler,  the  third 
J)th  row  of  the 

Irows  and  col- 
equal  deter- 

imn. 


25.  (I,  in  tlu!  first  row  and  fir^t  column. 

26.  6*3  in  the  second  row  and  fifth  column. 

27.  6'4  in  the  third  row  and  second  column. 


Given  A--|</,  h.,  r^  d^  c-^  J^  (j^\,  determine 


31.    A,,. 

35.        Ar 


28.    A,, 6.  29.    A,, 5.  30.    A,  t- 

32.    ^2,6-  33.    ^l.f,  5.  34.    -4 3^  7.  WW.    -(i5, 2- 

36.  Prove  that  a  determinant  will  not  be  changed  in  value 
by  any  permutation  of  the  rows  and  the  columns 
which  merely  changes  the  order  of  the  elements 
of  either  diagonal,  without  changing  the  elements 
themselves. 

§  59i  Theorem  III.  If  A  denote  any  determinant  of  order 
n,  £ik^^^  the  determinant  formed  from  A  hij  strilcinr/  out  the 
l)th  row  and  the  '\th  column,  ayid  ap^^  the  (p,  (\)th  clement 
of  ^,  then  ivill 

A  ^(-irK,^!.. -«•-',. ^2,,+ -{-{~ir-\„\„-\- ]. 

Cor.  1.   A  =  a,,,,.4,,,  +  a2,,A.,5  +  «;(,,^i,7+ 4-««,5-4„,, 

=  Op,  1-4^,  1  -f-  ttp,  2-4 j>,  2  T"  (^p<  3-4j),  3  r \   ^r,  n-4p,  n- 

Examples. 
1.    Let  A  -^--  '•  tti  h-i  c-i  I  and  ^  =  3. 


.-.A 


a,  A,   (\ 

Ci 

(h  Ih 

a-,  l>i  ('2 

-(-1)^ 

c,  a-i  h, 

«3     h     <?3 

C-i  c?3  Ih 

(-!)■'{' 

1 

f/2     h 

a-A  h^ 

— 

'Ci 

a^  hi 

-\-c. 

(h  Ih 

052     ^2 

} 


-(-l)Xr,A,,3-c•2A,,3  +  ^^,A3,3). 


326 


DETERMINANTS. 


2.    Let  A  ^  I  a,  h^  c-A  di  c^  j  luid  q     -  4. 

.-.  A  ^=  (—  !)•■'  I  di   «2  b-i  6'4  6'3  I 

=^^  (-  1)^   (di   I  a,   /A'.   ^'4   (-5  i     -  ^2  I  «1    ^3  C'4  ^5  I 

+  <^3  I  ^'l   ^^2  <^4,  ^5  I    —  <i>  I  «1   ^2  ^3  C'h  i 

+  C?5  I  <7,,    />.,   ^-3  <?4  I  ) 

-  (-l)'HA,,,  --r4A,.r|-t/,A3,4     </,A,.,  +  r4A,,0. 


Cor.  2.     7/"  /Ac  clcrnaits  of  the  \)th  row  all  vanish  exccj^t 
the  qth,  then  shall  A  =  (  -  l)'''^''ap,  ,jAp,q. 


a  b  0  c 

d  e  f  g 

h  k  0   I 

tn  n  0  2^ 


Example. 


=  (--l)H-/  d  e  g 

0  a,  h  c 

0  h  k   I 

0  7/i  n  I) 


( -  i)"y 


Cor.  3.  If  the  elements  on  one  side  of  the  pmicipal  diag- 
onal of  a  detcrminani  be  all  zero,  the  determinant  tvill  be 
equal  to  the  product  of  the  diagonal  elements. 

If  the  elements  on  one  side  of  the  secondary  diagonal  be 
all  zero,  the  determinant  will  be  equal  to  the  product  of  the 
secondary  diagonal  elements  multiplied  by  (—  l)5"("-^>,  n 
being  the  order  of  the  determinant. 


1. 


a  b  c 

—  a 

d  c 

0  d  c 

0/ 

00/ 

Examples. 
=  adf. 


2. 


00  0  a 

—  —a 

0  0b 

—  ~-ab 

0  d 

00  b  c 

Ode 

a  h 

Od  c  f 

g  h  k 

g  h  hi 

=  abdg. 


((, 

b 

c 

h 

k 

I 

7)1 

n 

P 

^4  Csl 


vanish  except 


+YI  a   h  r 
^  h  h  I 
m  n  p 


principcd  diag- 
minant  will  be 

ry  diagonal  he 
product  of  the 
(__  l)i"("-^>,  n 


\d 


=  abdg. 


DETERMINANTS. 


007 


Cor.  4.  The  order  of  a.  determinant  may  he  raised,  with- 
out altering  its  value  by  prefixing  a  colimin  of  zeros,  and 
superposing  a  row  of  elements,  the  fi,rst  of  which  must  be 
unify,  bat  the  others  may  he  any  finite  quantities  whatever. 


Example. 

ai- 
«2- 

-  X     hi  - 

-  X     hi  - 

1          X             y 
0     rr-i      X     hi  —  y 
0     aj-—  X     b^  -■'  y 

Theorem  IV.  If  each  clement  of  a  row  of  a  determi- 
nant consist  of  two  terms,  the  determinant  may  he  resolved 
into  the  sum  of  two  determinants,  the  first  of  which  is  got 
from  the  original  determinant  hy  striking  out  one  term  of 
each  of  the  elements  in  question,  and  the  secojid,  by  restoring 
these  and  striking  out  the  others. 

Conversely  :  The  sum  of  any  number  of  determinants 
which  are  alike,  except  as  regards  the  mth  roiu  in  each,  is 
equal  to  a  <  determinant  which  is  like  the  given  determinants 
except  that  each  element  of  its  mth  row  is  equal  to  the  sum 
of  the  corresponding  elements  of  all  the  givov  determinants. 


Examples. 


a-^-x  d  g 
h  —  y  e    h 

c-\-z  f   k 


a       0       c 

d       c       f 

g-\-m  h  k 


— ■ 

a  d  g 

x     d  g 

b  e    h 

~y  c   h 

cf   k 

z    f  k 

n 


a  h  c 
de  f 
g  h  k 

(I 


h 


d  e    f 


m 


0 


n 


3.    I  «!  hi  c-A  c/4 1  +  I  «i  ^2  c-i  r/i  I  -f- 
^|«i  ^2    (^3  +  63+^3)    d^ 


«i 


0-^ 


di 


328 


di<:termtnants. 


Theorem  V.  If  each  clement  oj  ayiy  row  oj a  determi- 
nant be  multiplied  (or  divided)  by  the  same  factor,  the 
detd^niinant  will  be  nibiltipUed  (or  divided)  by  the  said 
factor. 

Cor.  1.  If  all  the  elements  of  any  row  be  divisible  by  a 
common  factor,  such  common  factor  may  i  struck  out  of 
these  elements  arid  written  as  a  eocjjicieni  outside  the  bars  of 
the  resulting  determinant. 

Cor.  2.  If  the  siyn  of  every  element  of  a  row  be  chanqed, 
the  sign  of  the  determinant  ivill  be  changed. 


Examples. 


1. 


3  a  b  c 

--3 

a  b   c 

2>d  e  f 

dc  f 

Sg  h,k 

g  h  k 

2. 


9 

4 

8 

3 

6 

9 

5 

15 

20 

=  2.3.5 


1  2  4 
12  3 
1  3  4 


-  30  X  1  =  30. 


Here  the  common  factor  2  is  struck  out  of  the  first  row 
of  elements,  3  out  of  the  second  row,  and  5  out  of  the  third 
row,  and  their  product  is  written  as  coefficient  of  the  result- 
ing determinant. 


3. 


be  1  a 
ca  1  b 
ab  1  c 


abc 


abc  a  a^ 

1  a  a^ 

abc  b  P 



1  b  h' 

abc  c  c^ 

1  c   c'- 

4. 


0 


a 
b 


a     b 
0     c 
-c  0 

— —  ■■  — 

0 
a 
b  - 

a     b 
0 
-c  0 

'    (- 

ly 

-(- 

ly 

0  --( 
a     0 
b     c 

X  -b 
—  c 
0 

0 

—  a  b 

1 

a 

0     c 

I 

b 

c     0 

1 

DETERMINANTS. 


329 


of  a  ddernii- 

X  factor,   the 

by   the  said 

divisible  by  a 
struch  out  of 
Idc  the  bars  of 


iw 


be  channed, 


1  =  30. 


)f  the  first  row 
out  of  tlie  third 
nt  of  the  result- 


But  this  is  the  original  determinant,  say  A,  with  its 
columns  written  Ud  rows  ; 

.•.A  =  (-1)'A;   .-.  A  =  0. 

Theorem  VL  Any  determinant  can  always  be  trans- 
formed into  a  deierminant  of  the  same  order  in  which  the 
non-zei'o  elements  of  any  one  row  or  07ie  column  are  all  unity. 

Examples. 


1.   Let 


A  = 


«! 

b. 

Ci 

tta 

b. 

0-2 

tts 

b. 

C-2 

Multiply  each  element  of  the  first  column  by  biCi,  each 
element  of  the  second  cohimn  ))y  r^ai,  and  each  element  of 
the  third  column  by  aibi ; 


■.  a^b^Cx^ 


aibiCi  b^CiGx  Cittibi 
aJjxCi  b^Ciai  c^aibi 
a^bxCi     biCxax     c^axbx 


axbxCx 


.  A  = 


A' 
ttxbxCi 


1  1  1 

«2^i'^i     biCxax     c^axbx 
aJ)xCx     b-^Cxax     CsUxbx 


=  ai^i^iA',  say ; 


0  —a  b 

a     0     c 
b     c 


0 


2.    Reduce 


3-274 

-5437 

6      3      5-2 

4      6-35 


to  an  equivalent  determinant  having  the  elements  of  its 
second  column  all  unity. 

The  least  common  multiple  of  the  elements  of  the  second 
column  is  12,  and  the  quotients  of  12  by  these  elements  are 
—  6,  3,  4,  and  2,  respectively.     Multiply  the  first  row  by 


330 


DETERMINANTS. 


—  G,  the  second  by  3,  the  third  by  4,  and  the  fourth  by  12, 


and  divide  the  determinant  by 
Tlie  result  is 

1 
'  144 


-18  12  -42 

-15  12        9 

24  12       20 

8  12    --G 


24 

21 

-8 

10 


Gx3x4x2  =  -  144. 

1  I  -  18  1  -  42  -  -  24 

l^i-15  1         9      21 

"j      24  1       20      -  8 

I        8  1     -G       10 


Ex.  74. 

Expand  the  following  determinants : 

2. 


4. 


(I 

b 

c 

d  0 

c 

i.^' 

0 

h 

1 

0 

a? 

1 

0 

V' 

1 

f* 

& 

5. 


(I 

h 

c 

d 

e 

0 

9 

h 

0 

5  0  4 

3 

7 

3 

4 

0 

5 

3. 


6. 


X,  ?/,  -, 

0    0    z, 

^'3  y^  ^-i 

5  0  4 
0-2  0 
4      0    3 


a, 
0 

h,  0 

(^(■i 
h. 

0 
0 

<?2     Cs 

e.,  0 

0 

8. 


X 

y 

0 

0 

0 

X 

y 

0 

0  0 

X 

y 

y 

0  0 

X 

a 

0  e 

0 

X 

h 

0/ 

X 

k 

c 

0  X 

0 

0 

d 

X  g 

h 

I 

X 

0  0 

0 

0 

10. 


«1 

h^  Cx 

di  ei 

0 

h   C^l 

C?2    ^2 

0 

h  0 

d,  0 

0 

h,  0 

d,  0 

0 

h  0 

(4  er. 

Show  that 
11. 


12. 


a^h.O  0 

a2  h,  0  0 

tta   ^3   Xi  2/1 

«4  ^4  ''^2  y-i 


X 

-1 

0 

0 


«!     ^>'l 

X 

.'^1  yx 

tta   ^2 

X.,  2/2 

?/ 

0    0 

— 

X 

f    0 

-1 

^    2/' 

0 

-1    X 

X 

y 

0  0 

-2/ 
0 

0 

X 

-y 
0 

y    0 
^   y 

—  yx 

e  fourth  by  2, 
X  2  =  -  144. 


-42 

0 

20 


-24 

-8 
10 


•Ti  y,  2i 

0   0    Z.2 

Xz    ?/3    Zs 

5  0  4 
0-2  0 
4       0    3 


^  0 
0 

X 


Cx 


cL  c 


1  ^1 


0     6^3    0 

0   d,0 

0  (4  ^5 


DETERMINANTS. 


331 


Resolve  the  following    determinants    into    determinants 
with  monomial  elements : 


13. 


15. 


17. 


«!     ^1     !     1       ('i  j 

«.,  h,-\rx    r.,\ 
«3  hz  +  x^  r, 

((,  -  -x-\-y  h., 
ch  +  x  -y  b; 

^1 
^3 

Xi  —  a  yx  d^  - 
x.,--b  y,  b""- 
^•3  -  0   yz  c'  - 

-2^2 

14. 


16. 


18. 


(/,    h^    (\  -f-  OC^ 
(f.,    b-i    (\  -f-  X^ 
<f:s    b-^    C^  -f-  X 

(i,-\-x  bi-i-y 

<'i   -.^'  ^2  +  y 
a3-f-.T  ^3-  -y 

.r  4" «     <^^        y 

6?          /      z-\rk 

Combine  into  a  single  determinant 


19. 


Xx  yx  z, 
X'i,  0    z-i 

^h   y:j   23 


+ 


^'3  2/3 

23 

U-i   U2 

0 

^\  Vx 

Zl 

20. 


(ix 


^'l     ^1    I   + 


I    ^3 


«!  tta  a^ 

^1     63     ^4 
<^1     ^3      ^4 


21. 


.ri  — fti  ?/i  2;, 
^2  — «2  2/2  ^-.^ 
^3      ^3  y.'t  2;,{ 


+  I2/1  Zi  «i  — Wi 
!  y^  Z2  «2  —  W2 

iys    Z3    «3-'W3 


0    0 


y 


0 


^     2/ 
-y  X 


22. 


23. 


-^n-\-p     m-^-n—p   -\-     x-^y-\'Z      x  —  y-\- 


y 


y 


1)1  —  71 


P 


m 


—  n-\-p 


3  yx  zx 

V  yi  2i 

— 

4  y2  Z2 

6  y2  ^2 

5  2/3  23 

5   ys   2:j 

10 
10 
10 


2/1    2i 

2/3  Z3 


-10 


2/1 


-I 


-  10a;i  y2  —  x-i, 
5  ys 


"2 


a;, 


332 


DETERMINANTS. 


1,1 


Show  that : 


24. 


25. 


a  + 11  h  -\~v 


a^u  h  ~  V 
c  —  X  d-    y 


rti  hi  Ci 

-— 

((-i  h-i  c?2 

«3  ^h  c-i 

0  hi  ci 


+   <-/, 


-  9 


-Vh,, 


.h 

- 

^2 

u 

V 

d 

X  y 

0  ., 

+  ^3 

0  hi 

«3   <^3 

a.. 

0 

26. 


^'i  hi 

Ci  di 



ch  h., 

c,  d.i 

(J"A    h-A 

(\\  (h 

Hi  hi 

Ci  d^ 

0    hi  Cx  c/i 

+  «i 

^2  ^2  4 

a^  0    <?2  tZj 

^3     <?3     t4 

«3  ^3  0    c/3 

Z>4  Ci  di 

a*  &4  ^4  0 

+h 


0    ^1  di 

4-^3 

Cti  ^3   tZa 

^4   Ci   di 

0    ii  di 

+CZ4 

0    ^»i  ci 

rt2    0      t/2 

a^  0   (72 

0-4  64  (;?4 

ttj    Z>3    0 

III.   Evaluation. 

§  60.  Theorem  VII.  If  two  rows  of  a  determinant  he 
ide7itical,  the  determinant  will  he  equal  to  zero. 

Cor.  If  the  corresponding  elements  of  two  rows  of  a  deter- 
oyiinant  have  a  constant  ratio,  the  determinant  will  he  equal 
to  zero. 

Examples. 


Xi  yi  Xy   =0. 

x^  y<i  Xi 

^3  Vz  ^3 


mxi  3/1  nxi 

—  mn 

mx^  2/2  ^^2 

mx^  3/3  7iXz 

7nai  -{-nhi  «,  hi 
ma2-{-nh.2  a.2  h^ 
ma^-\-nh.i  a-^  Ih 


mai  ai  hi 

ma^  rtj  hi 

ma-i  a-i  h^ 

nhi  «i  hi 
nh-i  a.i  Z>2 
nhi  as  bs 


=  0. 


Theorem  VIII.  The  value  of  a  detominant  will  not  he 
altered  if  to  the  elements  of  any  roiv  there  he  added  equimul- 
tiples of  the  corresponding  elements  of  any  other  row. 


—  ij 

u 

V 

X  y 

I 

■\-C'i 

0   hi 

3 

a., 

u 

I. 

is 
i. 


0   c^ 
h,0 


leterminant  he 

•ows  of  a  deter- 
will  he  equal 


ax  hi 
tt'i  hi 
az  hi 


=  0. 


\ivt.  will  7iot  he 
tided  equimul- 


i     -V 


DETERMINANTS. 


333 


Examples. 


1. 


Ox  hx  Cx  1  — 
a-,  b.,  c, 
«3   b^   (?3  ! 

Ox    hx    Ox 

((■i  b^  C2 

«3    63    <?3 

mhx  hx 
inh.i  />2 
■mZ^s  b-s 

^3 

+ 

rti  -f  Tnhi  -j-  7iCx  hi  Cx 
a-i  +  'mbi  +  72^2  hi  c^ 

as  +  wis +  71^3    ^3    <^'3 

ncx 

/>,  i?» 

nc-i 

/>2    ^2 

??y?3 

^3  <^t 

2.    Evaluate 


12     3  4 

8     7     G  5 

1     3     6  10 

36  28  21  15 


From  the  elements  of  the  fourth  column  subtract  the 
corresponding  elements  of  the  third  column,  and  write  the 
remainders  as  the  corresponding  elements  of  a  new  fourth 
column.  Do  the  same  with  the^  second  column  instead  of 
the  third,  and  the  third  instead  of  the  fourth,  and  then 
with  the  first  and  second  columns  instead  of  the  third  and 
fourth.     The  resulting  determinant  will  he  : 

1111 

8  -1  -1  -1 

12      3      4 

36  -8  -7  -6 

In  this  determinant,  add  the  elements  of  the  first  row  to 
the  corresponding  elements  of  the  second,  and  also  ten 
times  the  elements  of  the  first  row  to  the  corresponding 
elements  of  the  fourth  row  ;  the  result  will  be  : 

0  .  . 


1111 

--9 

1  1  1 

9  0  0  0 

2  3  4 

12  3  4 

2  3  4 

46  2  3  4 

LT  row. 


by  Theor.  III.,  Cor.  2,  and  Theor.  VII.     Hence  the  given 
determinant  is  equal  to  zero. 


334 


DETERMINANTS. 


3.    Evaluate 


i  ::  ■!    s 

2  0  7   10 

5  9  :;    1 

7  3   1     0 

■ 

Take  twico  {]]{)  third  cnluinii  iVoiii  tlio  fourth  lor  a  new 
i'ourth,  twico  tin;  socoud  from  tho  third  for  a  new  third, 
and  twice  the  first  from  the  second  for  a  new  second  ;  the 
result  is  : 


1 

0 

0      0 

— ■ 

2 

-5 

-4 

2 

2 

5      4 

-1 

15 

5 

5 

-1 

15  -   5 

11 

1 

-11 

-i2      1 

To  the  second  row  add  the  third,  and  from  the  result 
subtract  the  first  row,  and  write  the  remainders  as  a  new 
second  row.  To  the  first  row  add  four  times  the  third,  and 
write  the  sums  as  a  new  first  row.     The  result  is : 


1  -  42  -- 13  0 

-  14-12  0 

"11    -2  1 


4. 


-42-131-  (-)n4 

-14      12 1 

14(24-1) -322. 


3  13 
1  12 


14 


2    1 
1  12 


1  1  1 

— 

a    h    c 

a'  h'  c' 

1 

a 
a' 


0 

h  —  a 
b'  -  a' 


0 

c  —  a 

c'      a' 

a) 

1 

a 

a'  I 

0 

1 


0 

1 


b-}-  a  c-\-a 
=  (b~  a)(c  -  a)[(c  +  a)-{b-{-  a)] 
=  (a  -  b)(b  -  e)(c  -  a). 

This  determinant  may  also  be  evaluated  thus : 

The  determinant  vanishes  for  a  ==■  Z> ;  therefore  a  —  b  is  a, 

factor  of  it.    By  symmetry,  b  —  c  and  c  —  a  are  also  factors. 

Now  the  determinant  is  of  the  third  degree  ;   there  are, 

therefore,   no  other  literal  factors   than  these  three ;    the 


f 


I 


DKTKRMINANTS. 


335 


irth  for  a  nnw 
r  a  lU'W  tliinl, 
2W  second  ;  i^»<' 


4 
5 
1 


from  the  result 
inders  as  a  now 
les  the  tliird,  and 
?sult  is : 

.131   .1412    11 

12  112| 


0 
1 

c-\-a 

\(h  +  a)] 


thus : 
lerefore  a  —  &  is  a 
la  are  also  factors, 
legree  ;  there  are, 
1  these  three;   thej 


d(»termlnant  thcrofore  ~7n(a  —  h){l)  -  r)(r  --  a),  whciciii  m 
is  numerical.  To  determine  vi ;  Tlic  principal  diagonal  is 
ba\  and  the  factors  give  7(11^^,  hence  ?/i --  + 1,  and  there- 
foi'c  the  determinant  is  ecpial  io  {a-  -  h){l>  (')((•  <<),  as 
was  otherwise  already  jtroved. 

6.  0  X  y  z    ~    X  -(•  y  -[-z  x  y  z 

x^-y-Vz  0  z  y 
x-\-y  -\  z  z  0  X 
X'\-y-\~z  y  X  0 

which  shows  that  x  -f-  y  -\^  ^  i«  a  factoi'. 

Multiply  the  first  and  fourth  columns,  and  the  second 
a!id  third  rows,  each  hy  ~  1,  which  is  equivalent  to  multi- 
plying the  determinant  by  (  -  1)*  =  1 ; 


0  X  y  z 

-.— 

X  0  2  y 

y  z  0  X 

z  y  X  0 

0 

X 

y 


X 

0 
V 


V 

—  z 

z 
0 

y 

X 

X 

0 

and  taking  the  sum  of  the  columns,  as  before,  for  a  new  first 
column,  x-{-y  —  z  is  seen  to  be  a  factor.  Similarly,  x  —  y-{-z 
and  —x-\-y-\-z  may  be  shown  to  be  factors.  The  deter- 
minant is  of  the  fourth  degree,  and  four  linear  factors  have 
been  found; 

.-.  A  =  ?7l(a:  +  ^/  +  z)(2/  +  ^-a:)(c  +  ^-3/)(^+y-2)• 
The  secondary   diagonal  is  +2*,   and   the    factors  give 
—  mz^^  tn  -~  — 1. 

.•.A  =  -~(x-i-y-{-z)(y  +  z-x)(z-^x~2/)(x-\-y-z). 


6. 


a 


+  1      ab 


ah 
ac 
ad 


ac         ad 
b'  +1      be         bd 
be      c'^  +1      cd 

bd        cd  d'  +  1 


=  A,  say. 


330 


DETERMINANTS. 


Multiply  tlio  first  rolumn  by  <i,  nud  then  strike  out  of 
the  first  row  the  common  factor  a  ;  this  will  not  chanf^'e  tlie 
value  of  the  determinant,  which  denote  hy  A. 


a'+l        b  c  d 

d'b  h'  +  1  be  bd 

((?c          bo.  (i^  -}-  1  cd 

(i\l         bd  cd  d'  4-  1 


Similarly,  operate  with  b  on  the  second  column  and  sec- 
ond row,  with  c  on  the  third  column  and  third  row,  and 
with  d  on  the  fourth  column  and  fourth  row ;  then 


a'  +  1      b"" 

&         d" 

a'       b'  + 1 

c"         d^ 

a'          h' 

c'-\-l       d} 

a'          h" 

&      d'  -f 

Take  the  sum  of  the  columns  for  a  new  first  column,  and 
write  the  common  factor  outside  the  bars. 


A --=(«'' +  ^»2  +  c»  +  c^»+l) 


7. 


Let 


=  aH y  -^-^^  r.r  +  1.   (See  Theor.  Ill,  Cor  8.) 

X  y  z    ^^   X -\-y -\-z  y  z 

A—    z  X  y         x-\-y  -{-  z  X  y 

y  z   X  x-\-y-\-z  z   X 

'.  X  -\-  y  +  2  is  a  factor  of  A. 

'^  -f- 1*>  +  1  —  0)  ai^^^  .".  o)^  =  1.     Multiply  the  second 


column  by  w  and  the  third  by  w'^,  the  second  row  by  w'^  and 


\       y          c^          d^ 

1  h'-\-\    c?       d"- 

1      h'      c^+l      d^ 

1     h'       c"    d'-^-i 

f  the  others  ;                         I 

1  b-"  &  d^ 
0  10   0 
0  0   10 
0  0   0   1 

triko  out  of 
t  rhan<'o  the 


MTKUMINANTK. 


3;r 


imn  and  sec- 
lird  row,  and 
then 


it  column ,  and 

c'         d'     1 
c'         d' 
c^  +  1      d' 
c'      d'  +  l 

era; 

d' 
0 
0 
1 

or.  ] 

II.,  Cor  3.) 

[ply  the  second 
row  by  w^  and 


the  third  hy  o>,  which  is  c<|iiiviilciil  to  iniiltiplyiii^  A  hy  <«", 
which  ^  1. 

X     «.)//    (i)'2     =    X-\~  Oil/    \'  0)^Z   0)1/    ut'z 
.'.il--     iiii'z     X     01//  \  X -\~  Mf/ '\- ot'z     ./•     O)// 

0)1/    oi^Z     .1'  \  M -l- 0)t/ -\- 0}^Z   w'^Z     X 

.'.  X  -{-  wy  -h  w'z  i«  a  factor  of  A. 

Oporatc  with  o>'  in.shiiid  of  o>,  arul  tlicroforo  with  <.)  in 
steud  of  0)',  and  x  -j-  w'y  +  <"2  will  bo  seen  to  be  a  factor 

of  A. 

.'.  A  =  ??i  (.r  -f  y  +  z)  (a;  -f-  <"y  4-  '^'^z)  f-^'  -f  '^V  {•  w-) 

in  which  ni  is  numerical.     The  principal  diagonal  of  A  is 
x^,  and  the  factors  give  i/ix^, 

.•.  7/t    -  +  1. 

.-.A  -=  (a;  -f-  7/  -f  2)  (a:  +  a>y  -f  w'z)  (a:  -|-  <o'y  +  ws) 
=  a;^  +  y' +  2^  —  3a;?/2. 


Ex.  75. 


Evaluate 


1. 


3. 


5. 


13-5      8 

4     7      2-6 

3  10    12      6 

-9     1     13     19 

2. 

1   -1 
1   -1 
1        1    - 
1        1 

1 
1 
1 

1 

1 

1 

1 

-1 

5        9        5-5 
9       15      19      23 
5       19       10       15 
5       23       15       25 

4. 

1  14  15     4 
8  11  10     5 

12  7     6     9 

13  2     3  16 

17  24     1     8  15 

23    5     7  14  16 

4    6  13  20  22 

10  12  19  21     3 

11  18  25     2     9 

6. 

1111 
abed 
a'  b'  c'  d' 
a'  b'  c'  d' 

838 


7. 


9. 


11. 


13. 


1 

1    1 

1 

a 

b    c 

d 

a' 

b''  c" 

(P 

a' 

b'  & 

cV 

1    1 

1 

1 

.c  /r 

& 

d'' 

tc'  //' 

c" 

d' 

a'  b' 

c' 

d' 

{ai-by 

a' 
b' 

(b+c) 

a    b  c 

0 

0    a  b 

c 

a'  b'  c' 

0 

0    a'b' 

c' 

DETERMINANTS 


8. 


10. 


12. 


a' 

{c-\-ay 


14. 


rs. 

1111 

abed 

a^  b'  c'  d' 

a*  b*  c'  d' 

0  111 

1  0    a:"  b' 

1     O^    0      6'^ 

1  b'  c'  0 

bi/-\-cz       bx  ex 

ay      ez-\-ax       ey 
az         bz      ax-^by 


X  a  b  e 

e  X  a  b 

b  e  X  a 

a  b  e  X 


15. 


X 

a 

b 

e 

d 

d 

X 

a 

b 

e 

e 

d 

X 

a 

b 

b 

c 

d 

X 

a 

a 

b 

e 

d 

X 

16. 


X  y  y  y  y 

y  X  y  y  y 

y  y  X  y  y 

y  y  y  X  y 

y  y  y  y  X 

17. 


X 

X 

y 

X 

y  y  y 

X 

y 

X 

X 

X 

y 

X 

y  y 

18, 


{b  +  e)'       e^ 
c'        {c+af 
h"  a' 


b' 

a^ 

{ai-by 


19. 


20. 


a -\- b  -{-  e -\-  d 
a  —  b  —  e-\-  d 
a  ~  b  ~\-  c  —  d 


U'+c+dy 


a.  —-  b  -  e  -\-  d 
a  -{-  b  -\-  e -\-  d 
a^b  —  c  —  d 


a 

a' 

a^ 


b' 

{e  +  d^oy 

b' 

b' 


a  -b  Ar  c  —  d 
a  ■\-  b  —  e  —  d 
a -{■  b  -\-  e -{•  d 


{d-\-a-\-by 


d-" 

d' 

d' 

{a-\-b-\-ey 


or, 
ran; 


DETERMINANTS. 


:v6d 


;  ex      I 

ax       cy 
I      ax\hy 


V 

X 


a 


a 


(«+^y 


h-c-d 

h^-c  +  d 


§  61.  Any  determinant  of  the  third  order  may  readily 
be  evaluated  by  the  following  method,  called  The  Method  of 
Sarms.     Let  the  determinant  be 


^1 

Ih 

Cx 

a., 

b. 

C'l 

Ch 

b. 

^'3 

Repeat  in  order  the  first  and  second  rows  below  the 
determinant  (or  the  first  and  second  columns  to  the  left  of 
it)  ;   thus, 


or,     (ix     ^i     Ci     r/i     bi 
\,  X     X     /, 
«2     f^     Ci     a^     ^2 
/,  X     X     \, 

«3        O3        CZ        «3        O3 


«!  b^  Cy 

a-i     O2     Ci 

X,  X 

«3       O3      C:i 
X,   X 

«1  ©1  Cy 

fin  On  Cn 


Form  the  product  of  the  three  elements  in  the  principal 
diagonal,  and  also  of  the  three  in  each  of  the  two  lines 
immediately  following  the  principal  diagonal,  and  parallel 
to  it.     In  this  case,  these  products  are  : 

chb-iC^,  a-ibiCi,  a^ibiCi ;  (or  chb^c^,  b^c^a^,  c^a^b^). 

Next,  form  the  product  of  the  three  elements  in  the 
secondary  diagonal,  and  also  of  the  three  in  each  of  the 
two  lines  immediately  following  the  secondary  diagonal, 
and  parallel  to  it.     In  this  case,  these  products  are  : 

«;A^l,    «2^i'"3,    aJhC'i  ;    (or  cJha^,    CliC'A,    ^iC^/'a)- 

From  the  sum  of  the  former  three  products  subtract  the 
sum  of  the  latter  three  ,  giving  in  this  case  : 

«/>'2^3  +  ciJhCi  +  aAc-i  —  (^3^2^^^!  +  chbiCi  4-  aib:iC.i), 
or,  taking  the  products  derived  from  the  right-hand  ar- 
rangement as  given  above  : 

«i^2^3  +  bic^a^  +  Cia.ib.^  -    {c^biCi^  -f-  tiic/,.,  -f  b^a^c-i). 
The  given  determinant  is  equal  to  either  of  these  expres- 


340 


DETERMINANTS. 


we 


sioiis,  which  are  of  the  same  vahie,  as  may  easily  be  seen, 
for  they  differ  merely  in  the  order  of  their  terms,  and  in 
the  order  of  the  factors  of  those  terms. 

In  practice,  it  will  soon  be  found  sufficient  merely  to 
imagine  the  rows  (or  the  columns)  rej^eated. 

§  62.  The  following  theorem,  which  is  an  immediate 
consequence  of  Theor.  VIII.  and  Cor.  2,  Theor.  III.,  often 
affords  the  quickest  and  readiest  means  of  evaluating  a 
determinant  with  numerical  elements. 

So  arrange  the  given  determinant  that  none  of  the  elements 
bnn  Cn„  ,  .  ,  km  shall  be  zero,  and  thai  all  the  elements  of  the 
mth  7VIU  after  1^  shall  be  zero,  then 

ai  bx  c'l  . . :  li  . . .  Si 
a.2  b-i  ^2 . . .  4  • . .  S2 
a-i  hi  ^3 .  .  .  4  .  . .  S;, 


^n    ^H    ^'n 


'-'mt/n.  .    .   .  fV, 


«1 

a„^ 


i  <^m 


a, 
a 


b, 

b,„ 

h, 

br,, 

•       • 

bra 
b,n 


m+1  ^m+1 


m 


b. 


'n 


bi      c, 

b,a        <^m 

>    •    •    • 

bra        C„, 

1   f^m 

bra-1  Cm-i 

)    •    •    • 

Ifa-l 

bra        Orr, 

'•••17, 

tm-\-\ 

l' 

bra        Cra 
bn         Cr, 

,   .   .   . 

4. 

.  s. 


.  6- 


ta-1 


'm-j-l 


Sn 


If  m—  1,  the  elements  of  the  first  row  will  be 


P2, 


This  method  of  evaluation  is  known  as  Oondensation. 


«i  bi 

bx  Ci 

ki  li 

a<i  b^ 

b^  C2 

'  •  ■  •    /•    / 

DETERMINANTS. 


341 


uly  be  seen, 
erms,  and  in 

it  merely  to 


iXi   immediate 

>or.  Ill-,  often 

evaluating   a 

of  the  elements 
elements  of  the 


'■in 

L 

rill  be 


Si. 


Sn 


Jondensation. 


Evaluate 


Examples. 


1  2  3 

4  1  2 

3  0  1 

0  2  1 


6 
0 
4 
1 


Here,  since  none  of  the  inner  elements  of  the  first  row  is 
zero,  we  may  take  m=  1 ;  then,  operating  on  the  first  two 
rows,  we  mentally  evaluate 


1  2 

2  3 

3 

G 

4  1 

1  2 

2 

0 

and  write  the  results,  which  are  —  7,  1,  12,  for  the  first  row 
of  the  new  determinant ;  similarly,  we  proceed  with  the 
first  and  third  rows,  and  then  with  the  first  and  fourth 
rows.  This  gives  a  determinant  of  the  third  order,  which 
we  divide  by  6,  the  product  of  2  and  3,  the  inne7'  elements 
in  the  first  row  of  A,  and  we  thus  obtain 


A  =  4- 


t 
-6 

2 


1 

12 

18 

— 

4 

9 

1 

7 
3 

9 


1  4 
1  3 
4     3 


This  determinant  may  be  evaluated  by  the  Method  of 
Sarrus,  or  the  condensation  may  be  repeated.  Condensa- 
tion gives 

-4  -1 


26     19 


=^.-76 +  26  =  -50. 


Ex.  76. 


Evaluate  by  the  Method  of  Sarrus  : 


1. 


1  1  1 

1  2  4 
1  3  9 


2. 


12  3 

2  3  1 

3  1  2 


3. 


2  -1 
1  2 
1       1 


1 
1 

2 


342 


DETERMINANTS. 


4. 


7 
1 
3 


6 
2 
1  - 


5. 


0  17 
21  0 
13  -  2 


Evaluate  by  condensation 


7. 


9. 


1 

-1 

0 

] 

2 

0 

3 

2 

0 

3 

1 

0 

3 

-1 

4 

2 
1 
3 
0 


0 
3 
1 
3 


0  1 

1  -2 
7  0 
1  -5 


11. 


8. 


10. 


0 

1 

1 

1 

0 

1 

1  - 

-1 

0 

1 

1  - 

-1 

1 

1 

1 

3 
4 
0 


6. 

1    1    1 
H  ^   i 

iii 

iil 

0 

1 

1 

8 

5 

0 

1 

1 

4 

1 

0 

1 

3 

1 

1 

0 

1  1  0 

2  2  1 

3  2  3 
5  4  2 


1  -1 

1  1 

1  1 

0  1 

1  0 


1 
0 
1 
3 


IV.  Multiplication. 

§  63.  Theorem  IX.  The  prod^ict  of  auo  dctcrrninanls  ^ , 
and  Aa  of  the  same  order,  is  a  determinant  such  that  the 
clement  in  its  -pth  row  and  qth  cohimn  is  the  sum  of  the 
2)roducis  of  the  elements  of  the  "pth  row  of  Ai  each  7nulti2olied 
into  the  corresponding  element  of  the  (\th  column  of  A^.     • 

Writing  a^,,  Xp^,  and  Ap^,  for  the  (p,  q)th.  element  of  Aj, 
A^,  and  of  their  product  resjDectively,  then 

-^pq  ^^  (^p,  \^l,q     r  ^'i>,  2  ^2,  q  "1"  ^p,  3  ^3,p  "T  ^tC. 

Before  forming  the  product  as  above,  Ai  or  A^^  may  either 
or  both  of  them  be  transformed  by  rearranging  the  rows  or 
the  columns,  or  by  changing  rows  into  columns.  The  pro- 
duct of  the  same  two  determinants  will  therefore  appear 


3. 


DETEKMINANTS. 


343 


8 
1 
1 
0 

0 
1 
3 


under  dilterent  forms  depending  on  the  arrangement  of  its 
factor-determinants,  but  these  forms  will  all  have  the  sam(3 
value.  If  one  of  the  determinants  to  be  multiplied  together 
be  of  a  lower  order  than  the  other,  its  order  must  be  raised 
to  that  of  the  other.     (Cor.  4,  Theor.  III.) 


1. 


a,  />, 

«2    K 


X 


EXAMl'LEH. 


■^•■2  y-i , 

rr-i   />,  ' 


X    .r,  .r, 

I  //i  y^ 


X 

Xx  x-i 

yi  y-i 

I 

X 

^1  yi 

^2    2/2 

1 

UyXi  +  ^,a,-2,  «,7/i  +  />»,y, 
a.^Xi  -f-  ia-2,  a2?/i  -f  ^sy: 

chXi  +  b^yu  ciix.,~\rh0. 
(iiXy  +  <7,y„  rt,a;2  +  a,y2 

a^xy  +  «2^\m  <''iyi  +  (^^y-i 

byXy  -y  b.,x.,,  b^T/i  +  b.,y., 


2. 


determinants  ^ , 

the  sum  of  the 
each  multiplied 

'Win  of  Ai.    • 
element  of  ^i, 

-|-  etc. 

)r  J^-i  may  either 
cring  the  rows  or 
Tmns.  The  pro- 
Ihcrefore  appear 


3. 


«!  bi'Ci 
^2  b-i  c.^ 
a-i  hs  t'a 


X 


Xi 

U/2 

X., 

yi 

y^ 

2/3 

^1 

Zt 

z-i 

a-iX^  +  ^..yi  +  ^22:1    <^'2-^'2  +  ''^2y2  ^-'  f^'i"-i    a.,x-,  +  L,y.,  +  c.,z^ 


CLx  hi  c'l   (/i  !  X  I  .^1  x.i 


«2     ^2     <^'2     <^2 


a.  ^u 


?, 


^3     ''3    ^3     ^'^S 

a^  ^1  Ci  (h 


?/!  y-^ 


(7i   hi   Ci   di 

X 

a^  Ih  <^2  d-i 

«3     ^a     <?3     C/;, 

«4     ^4     ^'l     <:4 

j 
1 

Xi    X., 

.r, 

X, 

y\  y-i 
0  0 

.^3 

1 

y* 

Z4 

0  0 

0 

1 

^i.Ti+Z^iy,  aiX-i-\-bxlJ'i  "i^'i+^iys  ft'i  "i^'4+/>iy4+^'i2:4+^A 

a,,.ri+/;2yi  <i'iX'i\-h-iy-i  a.iX.y\-h.,y.,^-\-(\,  (hXi-\-h.^y^\-i\Zi\~d.i 

a-iXi-^h,^yx  ((,;X.i-\\y,  ..■yVi-i-h^-^  \  c-.i  (-hXc\-h:syi\-c-iZ^-[(h 

aiXxr\-b^yi  a^x.^+hiy.,  a^x■i+b^yi~yc^  a^Xi+h,y,-\-c\Zi'\-d^ 


Here  x^,  y^,  Xi,  yi,  z^  are  wholly  arbitrary,  and  may  be 
lade  all  zero. 


344 


T)KTKRMINANTS. 


4. 


((x 


ch 


^- 


(h  ^h  ^'i  d^ 

X 

ct'i  hi  c\  cli 

«3     h    t';,     (li 

tti  b,  Ci  d, 

K 

h. 

ill 

~(h  - 

d. 

d^ 

Cx 

Ci 

h 

K 

a-. 

—  ((i 

d. 

d, 

-^3 

Cx 

\(hhi\-\\c.idi 

«:.  hi  l-l-l  C^  di 


0  I  «i  h  l+l  ''i  d-i  I  I  (ii  //,  l+l  r,  d;  I  I  cii  b^  \-[-\  c\  d^ 

0  I  «2  h  l+l  ^'2  (^3 1  I  fh  ^4 1  -fl  c-i  di 

«3  ^^2  l  +  l  ^3  ^^:!  II  0  I  ag  ^4  l  +  l  <"3  <'^4 

|«4  hi  l  +  l  6?4  cZi  I    I  ^4  ^>.,  l  +  l  <?4  c/a  II  a*  ^3  l  +  l  ^4  '-^  I  0 

This  is  a  skew  symmetrical  determinant  for 

(I  ^^1  h.,  \  +  \t\  d~i  1)  +  (\a.,  hil  +  l  e,  di  |)  =  0, 

by  Theor.  II. ;  and  the  same  holds  for  every  other  pair  of 
conjugates. 

§  64.  If  from  A,  a  determinant  of  order  n,  there  be 
erased  vi  rows  and  on  columns,  the  determinant  formed 
from  the  remaining  rows  and  columns  taken  in  order,  is 
called  a  Minor  of  A  of  order  9n  —  n.  The  minors  obtained 
by  erasing  one  row  and  one  column  of  any  determinant  are 
called  the  Principal  Minors  of  that  determinant. 

Two  minors  which  are  so  related  that  the  rows  and  col- 
umns erased  in  forming  one  of  them  are  exactly  those  not 
erased  in  forming  the  other,  are  called  Complementary  Minors. 

Thus,  I  ill  h-i  C3 1,      I  «2  h;^  f?4  |,     I  ^1  C4  ^5  I 

are  third-order  minors  of  |  «!  h^  c^  o?4  e^\,  and  their  comple- 
mentaries  are  the  second-order  minors, 

I  "4   <?5  |>         I  <^1   ^5  |)         I  ^*^2   <^3  |> 

respectively. 

I  ^1  C4  I   and   I  c~i  ilr^  I 

are  second-order  minors  of  |  rtj  h.,  c-^  d^  Cr,/^  \,  and  their  com 
plementaries  are  the  fourth-order  minors, 

I  ch  C3  d^fe  I  and  |  Ui  b^  c^f^  \ 
respectively, 


mi] 


Wll( 


DETERMINANTS. 


345 


7;,       ^3       ^* ' 

■  rta  —  Ota  •—  <^^* 
d2      d,       cU 

-  C^    —  C-s    —   ^4 

M  0 


for 

cy  otlier  pair  of 

rder  n,  there  be 

erminant  formed 

aken  in  order,  is 

minors  obtained 

determinant  are 

ant. 

he  rows  and  col- 
exactly  tbose  not 
plementary  Minors. 

tnd  tbeir  comple- 


re  I 


and  their  com 


The  principal  minors  of  |  <iy  b.^  C;^  d^  ^v,  |  are 

I  h  ^3  d^  Cr,  \,      I  hi  C3  di  ^5 1,      I  />,  r.^  J,,  <?5  I,   ... 

I  «2   ^3  f?4   ^5  I.         I  «1   <"3   ^^   ^5   |.     •    •    • 

complementary  to  rti,  a^,  a^,  •  .  .  ^i,  h-i,  .  .  .,  respectively. 

Hence,  if  in  any  determinant,  A^,^^  denote  tlie  comple- 
ment of  «^,,  (page  000),  (— ly^'ylp,,  will  be  the  princi^jal 
minor  complementary  to  a^,,. 

Theorem  X.  If  any  m  roms  of  a  dctcrtninant  he  selected, 
(Did  every  possihle  rtiinor  of  the  mth  order  he  forined  from 
tliein,  and  if  each  he  "tmiUiplied  hy  its  complementary  and 
(lie  p)roduet  affected  ivith  -\-  or  — ,  according  as  the  sum  of 
the  numhcrs  indicating  the  rows  and  the  columns  from  which 
the  minor  is  formed  he  even  or  odd,  the  sum  of  these  jyvodzicts 
IV ill  he  equal  to  the  original  determinant. 

Thns,  the  first  two  rows  of  |  Ui  h^  c-^  d^  \  give  the  six 
minors, 

I  a\  h.,  |,     I  nti  c.,  |,     I  «i  c4  |,     |  h^  c,  |,     |  h^  d^  |,     |  <?i  d.^  |, 

whose  complementaries  are 

I  Ca  di  |,     I  Ih  d^  I,     I  h-i  C4  |,     I  a^  di  |,     |  ^3  c^  |,     |  «3  h^  |, 

and  the  sums  of  the  numbers  indicating  the  rows  and  the 
•olumns  from  which  the  first  six  minors  are  formed,  are 

(1  +  2+1  +  2),     (1  +  2+1  +  3),     (l  +  2+l  +  4\ 
(1  +  2  +  2  +  3),     (1  +  2  +  2  +  4),     (1  +  2  +  3  +  4), 

".  I  a,,  h-i  C'i  d^  \  ~-=  \  «!  h.2^  I     I  (?;,  fZ4  I  -—  I  «!  C2'\     \  h.^^  d^  \ 
+  I  rti  d.  I     I  h-i  6^4  I  +  I  ^1  ^2  I     I  «3  ^^^4  I 


—  I  Z»i  rZj  I     I  (^3  ^4  I  +  I  ^1  d', 


a-i  A4 


340 


1)KTKRMINANTS. 


Similarly,  the   second,  third,  and    fitth    culuinna    being 
those  selected, 

I  c/i  />2  c'a  c/4  C5 


/>,    a,    T;, 

(t,  (I, 

— 

hi  c,  r^ 

«3    dr, 

bi    C,    Cr, 

(h  di 

"f~ 

by  c^  c^ 

'-h    t/5 

— 

hi  c^  c^ 

a-2  di 

-[- 

^'\    ^1   «^5 

a.i  di 

— 

1 2  (-'a  Ci 

a  I  (4 

^ 

b;    C,    fr. 

1  Ui  di 

'■'.i  c^  l\ 

a  I  d-i 

-|- 

h  Ci  <\ 

a  I  (4 

Ex.  77. 

Perform    the    following   multiplications,   (expressing  the 


results  in  determinant  form  : 


1. 


4. 


a.,  b., 


•r,  x-i 


2. 


o 
3 


5 
0 


3  -   G 
2       5 


3  5 

4  G 

5. 


X    V 

y  " 

2 

-1 


3. 


3!i 
1 


2  5 

3  4 


2 
3 


3  5 


5 
4  2 


6. 

2 
4  - 

3 

-2 

1  2 

2  4 

3  4 

3 

8 

-7 

7.    I 

2  2 

3  3 

f  f 

8     4 

a    y 

1 

3^ 

2 
7 
1 
9 

12     0     8 
3     T    <J 

2  3     4 
,   3     T    9 
112     1 

3  T   y 

8. 

h       b~-y 

a  +  y       h 

h      a  4^  y 

f 

9. 

a~y      h           (J 
h       h~y      f 

a  +  y       h          g 
h       b^y      f 

9       f    c-y 

9       I     ^  +  y 

10. 

1  a  a^ 
1  b   b' 
1  c   c' 

a"      h'      c' 

-  -  a  —  b  -  -  c 

1        1        1 

11. 

X   yi 

y  zi 

Z      X 

■ 

yi    X 

zi  y 

xi   z 

whe 

rein 

%'■= 

-1. 

21. 


DETERMINANTS. 


347 


Dlumns   being 


•4   '5 


«l   C?2    I 


expressing 


the 


3.    12  513  5 

3  4    4  2 


3!  \  -  2 


2     fi  8 

ij  T  9 

2  3  4. 

3  7  i) 

^     1  ^- 

^  T  y 


A         ^      1 

h-{-y     f    \ 

f     c-\-y\ 


12. 


13. 


16. 


17. 


19. 


20. 


iL  -  -  hi     -—  c  -{-  di 

X  - 

r  -  -  (h         a    -  hi 

u 

a  -f-  h      (I          h 

a  — 

0       h  +  c      h 

c          a      c  -{-  a 

X  +  yi      u  -}-  vi 
u  -\-  vi    X  —  yi 


>+i^       --ia  —Ih 

Ic       h-^-c-^-la       -~hh 
\c  —\a       c-\-a-\\h 


a  a 

a 

a 

a  h 

h 

h 

a  h 

a 

c 

a  h 

c 

d 

1 

a 
h 
c 


X 

h 


X 

c 

c     d 
d     e 


,2 


1  0 
1  1 
0-1 
0      0 


-  X 
d 

I 


0 

0 

15. 

X  y  z 

x'  z^  y' 

0 

0 

z   X  y 

y'  x'  z' 

1 

0 

y  z  X 

2'  y'  x' 

1 

1 

1 

X 

0 

0 

0 

1 

X 

0 

0 

0 

1 

X 

0 

0 

0 

1 

rti  -(-  a.^     (h  +  fh     ('3  +  <^(i 
hi  +  h.j,     h.,  4-  h^     h^  +  /;, 

<^l    +   <?2  ^2  4-   ^M 


«":<    ,-  Cl 


1 

1 

-1 

-1 

1 

1 

1 

-1 

1 

1- 

-2 

1 

1 

1 

1- 

-2 

1 

1 

1 

1 

-2 

9 

aU 

1 

1 

1 

«i+'^2+«3  h+hi+h^  ^iH-<?2+^3  di-}-d.r\-d3 

«2+«3+«4  ^2+^3+^4     ^2+<?3+^4   C?2+f4+'^4 

'^3H-«4+«l  ^3+^4+^1     C.i-{-Cc\-Ci    C?3-f  0?4-f  0?, 

a4+«rf-«2  ^-'4+^1+^2  ^4+^1+^2  di,-\-di-\-d2 


c,     0     - 
0     0 
0     0- 

h.i  (?.,  d-i 

c-i  d.i  0 

d^  0  0 

0  0  0 

0  0  0 

0  0  0 


h,     ■ 

Cx 

0    - 
1 

-Cx 
—  cZ, 

0 

0 

-1 


Cx 

0 

1 

0 


<7i 

0 

u 


Cx 

dx 

0 

0 

1 

0 


"x 

h, 

(U 


d, 

0 

0 

1 

0 

0 


^1 

hx 
C'l 

h. 


0 
0 


(ii  hi  Ci  di  0    0 

0  «i  hi  Ci  di  0 

0  0  (ti  hi  Ci   di 

rta  h.^  C2  d-i  0    0 

0  a-i  h-i  0-2  d.2  0 

0  0  «2  h.,  c-i  d<i 


1 
1 
1 


1x  x' 
2y  f 

2z    2^1 


'^^,2 

,.2 

lU^ 

!  u 

r 

w 

!l 

1 

1 

Fv  ;^ 


348 


DETERMINANTS. 


'><l    'XL' 


(( 


—  6* 


22.      1 

1  --8^  8A^ 

1  -3^   36'^ 

1  -?yd  3d'  -d\ 

and  (lodiico  therefrom  that 

9  (rt  -  -  h'f  {a  -  r)2  (a   -  dy  (h  -  r)2  (h 


a' 

//''    r'    d' 

•1 

rr 

h'   c'   d' 

rt 

h     C     d 

1 

1      1      1 

dy{c~dy 


~[(a-hy(c-dy-\  ih-dy{c-(iy-\-{a-dy{b  (^f. 


23.    If  y.. 


1 


1    a.> 


,1  ^  .1  tt  '"  I 


4-r/„'",  then  will 


(I. 


•2 


1 


(7.,       CI; 


(I'l 


n  •}. 


I      ''..    (/., 


« 


n   i 


5o     5i 


^4 


"St     '"^M-j-l    •''«  i : 


'n+1 


^n-l  '2 


.«tv 


:2n 


24.    If   A 


n, 


h, 


and  A'    -  I  A,  B,,  C,  ...  A', 


wluMvin   .1,,       A2,  A^,  ...     -/?,,    7>.^,  —Il;i,  otc,  ar(3 

the  pi'incipal  minors  of  A  ;  and  if  a,,  —a.^,  a;,,  ...  — ySi, 

Pi,      P.u  etc.,  are  the  principal  minors  of  A',  prove 

that 

AA'    ^A**  and  a^A^aiA",  a.,A-=«.,A",  etc. 

Note.  The  dotorminant  A'  is  called  tlio  Reciprocal  of  the  doter- 
Tiiiiiant  A  ;  and  tlio  olements  A^,  A.^,  ...  B^,  B.^,  etc.,  are  called  Inverse 
Elements  with  respect  to  a^,  a.^,  ...  h-^,  h.^,  etc. 

25.  Prove  that  a  minor  of  the  order  m,  formed  out  of  the 
inverse  constituents,  is  equal  to  the  complementary 
of  the  corresponding  minor  of  the  original  deter- 
minant multiplied  by  the  (??i  — l)th  power  of  that 
determinant. 

V.    Applications. 

§  65.    To  solve  the  simultaneous  linear  equations, 

a^x  +  h^ij  +  CiZ  ^  di,  (1) 

a.,x -[- h.,1/ i- c\z -^  d.^,  (2) 

chx -\- h,f/ -\- c.,z  ~- d,.  (3) 


DETERMINANTS. 


.'^40 


nil 


B.,  c, ...  a;  1 

-J^a,  ^tc,  aro 
-a.^,  a;,,  ...  —^1, 
'S  of  A',  prove 

^",   etc. 

'ocal  of  tlio  (loter- 
are  called  Inverse 

ned  out  of  the 
omplementary 
original  cleter- 
power  of  tliat 


lations, 


(1) 

(2) 
(3) 


Let  V  — "  I  fiAc,i\,  v«  ^A V.i I,  Vi.  '■  ■  I  (hfl2<^:i !,  Vc  I (^'Mil 
and  lot  Ai,  A^f  etc.,  denote  the  complementH  of  </,,  d.,,  etc., 
in  V- 

Multiply  (1)  by  A„  (2)  l)y  A,,  (.3)  hy  Ai,  and  add. 

Th(>refore.  by  Thcr.  HI..  0..r.  1,  iind  Theor.  VII., 

V.^•  ^  ^  V... 
By  using  7?,,  7>..,,  7?:,  instead  of  tI,,  ^.2,  yl^,  we  obtain 

V//  ^  ^^  Vfc, 
and  using  (\,  C^,  f^,  instead  of  Ai,  A.^,  A^,  gives 

V2;  ■--  Vc- 
The   method  here  exhibited   is  evidently  applicable  to 
the  case  of  71  linear  equations  containing  ?i  unknown  quan- 
tities. 

2.  Given  the  above  linear  equations  (1),  (2),  (3)  to  find 
the  value  of  ax  \-  fti/  -|-  y-- 

By  Theors.  VIII.  and  VII. 

ax  +  Pj/  -\-yz  a  fS  y     ~  0. 

a^x  4-  ^ly  4-  CiZ  cix  hi  c\ 

CLiX  +  ^.2?/  +  ^^2  (^2  ^2  ^2 

«3^'  +  ^32/  +  ^3-  ((i  h  ^3 

Therefore,  substituting  from  the  equations  (1),  (2),  (3), 

ax  +  /?//  +  y2     a      jS     y  \  ---  0  ; 
C?i  tti     hi     Ci 

di  a-i    hi    Ci 

^3  «3       ^3       ^3 

.-.   (cur  +  ^y  +  ys)  V  -  aVa  -  /3v6      yVc    ^  0 ; 
■••   OLX  +  ^y  +  yz  =  (aVa  +  /5V/-  ■ ;   yVc)  "-  V- 
This  result  necessarily  includes  the  solution  of  the  equa- 
tions found  above. 


350 


IjKTERMINANTS. 


3.  To  (Ictcrmiiif  the  coiKlitiuii  that  I  lie  liomogeru'ouM 
linear  e(|uatiuns 

axX'\-h^ij  I  6',z      0,  (1; 

a.,x  +  h.,y  +  c.,z  =■--  0,  (2) 

^^3.^'  -f-  %    f    <*:t2        0,  (3) 

may  coexist  for  values  of  .r,  y,  and  :;  otlier  tlian  zero. 
Multiply  (1)  by  yl,,  (12)  hy  /Ij,  and  (3j  hy  vl.,,  and  add. 

.'.  v^  =--  0, 

and  therefore  if  x  be  not  zero, 

v  =  o. 

4.  To  find  the  condition  that 

a,x'-\-h,x  \-c\^0  (1) 

and  Wi,x^  ■\h.p:-\-C2  —  0  (2) 

may  have  a  common  root. 

Multiply  each  of  the  given  equations  by  x ;  the  resulting 
equations  together  with  the  given  equations  constitute  the 
four  simultaneous  equations, 

ai.r'  +  hxX^  +  c^x  =  0, 

ctiX^  -f-  h^x  4-  ^1  =  0, 
a'iX^  +  h-i^x"^  -  f-  c^x  =  0, 

«2a;''' +  ^V'^+ ^2  ^^  0, 

which  are  to  be  satisfied  by  values  of  x\  x'^,  and  x  other 
than  zero ;  hence,  by  No.  3  above, 

tti  bi  6*1  0    —  0. 

0  ill  hi  Cx 

a^  Z>2  c-i  0 

0  a-i  b-i  c\ 


oniogonoous 

(1) 

(2) 
(3) 
1  zero. 

I3,  and  add. 


(1) 

(2) 


the  resulting 
constitute  the 


and  X  other 


nKTKinriN.xNT.s. 


351 


This  ifi  also  the  condition  tliat 

OxX^  -{-  bxX  -f-  r,  and  UxX^  -f"  ^^x^  H"  ^*i 
may  liavo  a  common  factor. 

5.    To  find  tlio  condition  that 

may  have  a  square  factor. 

Let  {x  —  7ny  be  tlie  square  factor.  Divide  tlie  given 
expression  ])y  x  —  ?«,  and  tlie  quotient  hy  x—  tn  ;  the  two 
remainders  thus  obtained  must  both  vanish.  These  re- 
mainders are 

a7ii^  -f  3  hn^  -f  3  nti  -f-  d 

and  ani^  -\-  2hni  -f-  c. 

.  • .   am'  -f-  8  Z>  //i'  -j-  3  rm  -\-  d^O, 
and  a7)i^  +  2  bf^i  -|-  c  —  0. 

Multiply  the  latter  equation  by  7n,  and  subtract  the 
product  from  the  foi'mer. 

.-.   1)7)1' +  2c7)i-\-d^0. 

Combining  this  equation  with  the  second  of  preceding, 
the  condition  required  is  found  to  be 


that 


an( 


ax' 


+  2hxi-c  =0 


lx^-i-2cx-\-d=0 


sliall  have  a  common  root,  and  the  condition   for  this  has 
been  found  in  No.  4  above;  viz., 


a 
0 


2h 
a 


0 


0. 


b  2c     do 

0    b    2c   d 


352 


DETERMINANTS. 


6.    To  find  the  condition  that  the  expression 

ax^  -f-  bif  -\-  cz^  -f  2/?/2:  -f  Icjzx  f-  2  hxy 
may  be  the  product  of  two  linear  factors. 

Let  the  factors  he  a^x  -f  p\y  +  yi^  and  a^x  +  ySa//  -f-  y.22:. 

Multiply  tii«so  together,  and  equate  the  coefHcients  of 
the  product  with  those  of  like  powei's  of  the  variables  in 
the  given  expression, 

2/=  /^ly-i  +  AVi        2  r/  =  a,y,  +  a,y,        2  A  =  a,^2  +  a-^^Si 
ttitto  +  a^tti      a,/?2  +  a.,^,      a.y.^  +  a^y, 

yitt2  -h  yatti     y,/?^  +  y>^i     yiy.j  -|-  y.yi 


A  /^  /• 

1 

'"  8 

1 

8 

ax    tt.2   0 

yi  y-i  0 


I  "2  ^2  y-i 
X  I  a,  ^,  y, 

iO   0    0 


0. 


Hence  the  required  condition  is  that 

((  h  (j !  ^  0. 
h  h  / 

9  I  ^■ 


7.    If 


and 


then  will 


y  =  a,  A^+ Ai^+ y.^ 

2-a,X+^,y+y;,;f 

ax?  +  />?/  -|-  <?z^  +  ^fyz  -f-  2^20:  +  2  7/^7/ 
=  ^  X^  +  7?F^  f-  CZ''  H-  2  i^y.^ 

+  2  6^^x+2^.ry, 


A    IT  G\^ 

a,    /?,    yi  ;" 

rr  li  (J 

II  B    F 

tt,  /3,  y, 

h  h  f 

0    F    C 

«:!    /5;(    yt  1 

9  J    ^ 

(1) 


(2) 


J 


y 

efficients  of 
variables  in 


-  a,ft  +  a^^i 

2  -I-  fty. 


0^ 


DETERMTNANTvS. 


1^53 


Substitute  for  :r,  ?/,  and  ;:  in  (2)  their  values  in  (1),  and 
equate  coefficients  of  like  powers  of  X,  1"  and  Z. 

B  ^  >^r  f  bp.'  +  ^ft^  +  2//?A  -I-  2:/ft|8,  -h  2  /./?,/?, 
6^  -  ayi^   h  />y/  +  ^y.'  +  2/y,y.,  +  2yy;,y,  -{-  2  Ay.y, 

+  ^(ftyi  +  )S,y,)  +  A(/3,y.  +  fty.) 

(7  ^  rtyitti  4-  by.^a.i  +  q/ga,  +/(y2a3  +  y;,a.^) 
+  9  (ysai  +  yi^a)  +  ^i  (yia-2  +  y'itti) 

^==  aai/?i  +  bafii  -\-  ca;,ft  -\-f(a,(S,  +  a,ft) 

+  ^(a,ft  4-  a.ft)  +  h(a,p,  +  a,ft). 

Now 


«!  /?!  yi 

a-i  /?2  y2 

«•{  ft  ys 


a  h  q 

hh-f 

9  J  « 


tt]    a2   ^1 

a  h  q 

«!  /?i  yi 

^i  ft  ft 

h  b'f 

02  fti  y2 

yi  y2  ys  i 

9  J   ^' 

aa  fti  ys 

aaj  -f-  /itta  +  ^tts    /itti  -  f-  /vtta  -f-  /a-,    9'a,  -|-  /aj  4"  ^^3  I 
«/8i+Aft+i/ft  /^/8r|-^ft+/ft  ?//?rf-/ft+^ft. 

«yi + /iy2 +i/y3  /^yi + ^y2  +yy,j  i/yi  +yy2  +  cy^ 


<(\  ft  yi 
«2  /?2  y2 

cia  ^3  y3 


Xl/ 


A 

H  G 

II  B 

F 

Cr 

F 

C 

(2) 


8.    Eliminate  x,  ?/,  and  z  from  the  e(iuations 


ax 


-f-  hf  +  ('2'  +  2fi/z  -\  -  2  ,jzx  -I-  2  /u;y  -  -  0,        ( )  ) 


hxX  -j-  /ly  +  ')>hz  —  0, 
Z\,:r  +  til/  -]-  m^iZ  -—  0. 


(3) 


35-i 


DETERMINANTS. 


First  Method.    Let  Aj  and  k^  ^^  liomogenous  linear  func- 
tions of  X,  y,  <and  z,  such  that  A.i(2)  +  X^  (3)  =  (1). 

.-.  ax^  +  hf  +  cz^  +  2fyz  +  2gzx  +  2 /a-y 

=  {ax  -f  •  %  +  <72;)  :r  +  (  A.r  +  hy  -\-fz)  y  -f  -  {<jx  -j-fy + t^z)  2 

=-  (k^x  -}-  /,v/  +  Wi?.)  Ai  +  (A'a^t'  +  4y  +  m^z)K 

•'•  ^^'<'  +  hy  +  ^3  =  X'lAi  +  h,\^,  (4) 

Aa:  +  Z»y  +  /z  ===  l^\^  +  4X3,  (5) 

^^  +  /y  +  ^2  =  ??ZiA,  +  wA2-  (6) 

Now  eliminate  .^',  y,  z,  Aj,  A2  from  (2),  (3),  (4),  (5),  (6)  by 
the  method  exhibited  in  No.  3,  page  350. 

a  h    cj  liy  lc<i     =  0. 

h  h  f  li  4 

g  f    c  m,x  7112 

ki  li    7)11  0  0 

k.2      I'i      7)l2      0  0 

Second  Method.     Multiply   (2)   by  x,  y,  and  z,  succes- 
sively, and  (3)  by  x  and  by  y. 

.'.  kxX^  +  7nizx  +  l\xy  =  0, 

Ay'         -{-Tn^z  ^]crxy  =  0, 

711]^  +  Ay2  +  ^\^^  =  0, 

kix"^  -j-  m^zx  +  l^xy  =  0, 

Now  eliminate  x\  y^  z'^,  yz,  zx,  xy  from  these  five  equa- 
tions and  (1),  by  the  method  of  No.  3,  page  350. 

a  h  G  2/  2y  2  A    =0. 

^1  0  0  0  mi  k 

0  /,  0  m,  0  h 

0  0  7n,  U  h  0 

TC2       0        0  0  7112        J; 


0     /.,    0 


71U 


0       h, 


DETERMINANTS. 


355 


inear  func- 

)• 

:+/y  +  Cz)2 
I  A,. 

(4) 
(5) 
(6) 

,  (5),  (6)  by 


id  z,  succes- 


[e  five  equa- 

b. 


9.    To  solve  the  simultaneous  equations 
UiX^  -f-  2  hxxy  +  hxif  ~  m^, 
citX^  -f"  2  h^xy  -]-  h.^y"^  -—  1112. 

Write  them  in  the  form 

{a,x  +  h,y)  X  +  {Ji^x  +  l\y)y  =  mi, 
(a,.r  +  'h{y)x  +  {h,x  +  hiy)y^--  m^. 


Let 


chxArfhV  hxx  +  h^y 


.'.  (v  + 1  ^i  ^2 1)^  + 1  ^1  'f^h  |y  =  o» 
—  1 «!  7/^2 1  '"^  +  (v  —  I  ^h  ^ti'i  I)  y  —  0. 

.'.   (v^  —  I  Ai  7^2  P)  +  I  «i  ^^2 1  I  ^i  m-i  I  =  0. 
.•.   V  =  =t  V(l  ^1  ^^'i  P  ~  I  ^1  ^^'^  I  I  ^1  ^'^2 1). 
Hence  v  i^^^y  bo  treated  as  known,  and  then  by  (3), 

_      (V+ I /ii  W-,  |):r 


(3) 
(4) 


y 


hy    'Tih 


Substitute  this  value  of  y  in  (1),  and  there  will  result  a 
pure  quadratic  in  x,  from  which  the  value  of  x  may  be 
immediately  obtained. 


10.    To  solve  the  simultaneous  equations 
aiX^  +  hxy"^  -\-  2  hixy  =  7?ti, 
a^x  +  h.^  =  m^. 


(1) 

(2) 


This  may  be  treated  as  a  particular  case  of  the  preced- 
ing, or  otherwise  as  follows  : 


356 


DETERMINANTS. 


Let 


Write  the  given  er|uations  in  the  form 

a-iX  -f-  S.^y  --  1712 

V    ^ -  (a^x  +  A,y)  5,  —  (/iio;  -] -  b^y)  a,. 

VU  cki^h  4-  («i^-  +  hi'!/)  ''^^^■• 

(V  f  /^I'/^^i)  -^  +  biiih^y  -  -  ^2'^??-, :     0, 

a-i  "     ^2  ^^^2 


^3) 


7/?2V  -  Al7??.2^+a2^2^?'l       0 
^2  Wn 


rtg'*???,!  —  0,???2'^ 


«j. 


0. 


wliieh  pure  quadratic  gives  at  once  the  two  vahies  of  v, 
which  may  consequently  be  treated  as  known.  Then  from 
(2)  and  (3), 

a^iX  +  £>2?/  =  m2, 

(ai^2  --  Jha<i)x  +  {JiA  —  iirta)  --  V ; 

two  linear  equations  from  which  to  find  x  and  y. 


11.    Eliminate  x  from  the  simultaneous  equations 

x^  —px^  -\-  qx  —  r  ~  0,  (1) 

y  =  ai  4-  ^lO;  4-  Ci.T^  (2) 


»'■•'•    "rf^-.  .' 


DETERMINANTS. 


.  1  r  »-• 

o5« 


(3) 


0, 
=  0. 


^1 

W2 


0  '  =  0. 


r) 


o 


i. 


0, 

values  of  V. 
Then  from 


Ay- 


luations 


(1) 

(2) 


Multiply  (2)  l)y  x,  and  in  the  result  substitute  the  value 
of  r*  given  by  (1). 


=  a^  +  ^^2-^  +  <?2'^■^  say. 
Repeat  with  (8)  and  (1)  instead  of  (2)  and  (1). 

=  «3  +  ^3^^'  +  ^■iX'',  say. 
Eliminate  a;  and  x'^  fi'um  (2),  (3),  and  (4). 


(3) 


«3  ^3       ^3  —  y 


=  0, 


which,  on  being  expanded,  gives  a  cubic  in  y. 

12.    To  fmd  the  condition  that 

U  ~  ax'  +  hx'  +  ex'  +  f/.r''  -i-  ex  +/ 
and  F  =  ax'  +  jS.i;'  +  72-  +  8 

may  have  a  common  factor,  and  to  find  that  factor,  apply 
the  method  of  elimination  exhibited  in  Exan  pie  4,  page 
350.     The  result  is  : 

If             a     b     c     d    e    f    0  0     =0, 

0     a     b     e     d    c    f  0 

0     0     a     b     c     d    c  f 

a     /?    y     8     0    0    0  0 

0     tt      i^     y     8     0     0  0 
00a/8y800 

000a^y8  0 

0      U     0     0     a     /tf    y  8 

t/"and  Fwill  have  a  common  factor  which,  to  a  constant 
multiplier,  will  be 


358 


DETERMINANTS. 

ax  +  b 
a 

C 
b 

d 

c 

c  J  0 
d    e     f 

ax-\-  /3 
a 

0 
0 

y 

a 

0 

8 

y 

u 

0  0  0 
8  0  0 
y  8  0 
(i     y     8 

If  this  determinant  vanish  identicaUy,  i.e.,  if  the  constant 
multiplier  be  zero,  f/and  F'will  have  a  common  quadratic 
factor  which,  except  as  to  a  constant  multiplier,  will  be 


ax'^  -\-  hx  -f-  <^'      d 

c 

/ 

a:<;^ -t- /3.H- y      8 

0 

0 

ax  -\-  (3     y 

8 

0 

a     li 

y 

8 

If  this  determinant  vanish  identically,  6^ and  T^  will  have 
a  common  cubic  factor  which  will  necessarily  be  For  V 
divided  by  a  constant. 

Example. 

Let  it  be  required  to  find  the  common  quadratic  fac- 
tor of 

Qx'-x'-ce'-]-  lOx"  +  14a;  -  40 

and  2x^-\  x^-x-\-lO. 

Followuig  the  above-described  method,  it  is  found  to  be 

-40 

0 

0 

10 


0 
10 


G^^ 

~x-l       10 

14 

2x'-^x-l       10 

0 

2x  +  1     - 1 

10 

2        1 

-1 

-10 

^x'-x-\-1 

14 

2x'-]-x-\ 

10 

2a;+l 

-1 

-100 


6a;^-3:p  +  6     15 
2a;' -f    x~l     10 


DETERMINANTS. 


oJ 


9 


the  constant 
on  quadratic 
r,  will  be 


td  V  will  have 
tly  be  V  or  V 


quadratic  fac- 


is 


found  to  be 


=  1500  I  t2x-     u;  +  2     1  j  -  1500  (2a;^- 3a:  +  5). 
I  :lx'-\-x~l     2  I 

Rejecting  from  this  the  constant  multiplier,  1500,  the 
common  factor  is  2.r'*^  —  3.r  +  5,  as  may  be  proved  eitlier 
by  actual  division  or  by  evaluation  of  the  determinant  for 
a  linear  factor, 

Ex.  78. 

Apply  determinants  to  solve  the  following  equations  : 

1.    3x-\-7i/  =  8, 
4a;  +  9y=ll. 

3.  3.T  — 5?/  +  4z  =  5, 
7x-\-2y-Sz  =  2, 
4a:  +  3?/—    2  =  7. 

5.  |(:r  +  27/)-i(3y 
x-\-7/  —  z  =126. 


2. 

<-i  X 

5?/- 

-20, 

Sx- 

4y- 

r-7. 

4. 

X 

-y 

z  = 

=  6, 

Ix- 

9iy 

+  llz=-- 

^64, 

23x- 

21  y 

+  24z- 

^54 

Az)^ii6z-\-bx), 


6.    l  +  lw  +  Ja;  +  Jy  +  i2  =  0, 

i  +  i^^  +  i^  +  iy  +  i^-O, 

i  +  iu  +  ix-i-}7/  +  \z=^0, 
24  15 


7. 


2x  +  Sy  3:r  +  4z 

30        ■  37^_ 

Sx  -{-  42  5?/  +  92 

222  8 


=  2, 
=  3, 
5. 


5yH-92      2a;  +  3y 

8.   115(13  -  x)  +  719  (y  -  19)  -  590(37  -z)  =  27, 
5(13-x)+2_37-2._^ 


y 


-19 


y 


-21 


3G0 


DETERMINANTS. 


9.    u-\-h^ 

a?-j-6  = 

y-\-b-- 

z^b- 

a(x-}  I/),  10.  {(I  \  b  -j-  c')x 

(a+l)(y+2),  (a+d-\-c)y 
(a  +  2)(2  +  w),  (b  +  d+f)z 
{a-\-S){u-\-x').       Generalize. 


mj-\  bz  4-c, 
ax  -\-clz  -j-Cy 
bx-\-dy-\-S' 


11.  Given  Xy  =  b,y^,  y2  =  cixXy-[- y„  x.,-=^h,y.,-\- x^, 
2/3  =  «2^2  +  y2,  X3  =  b:iyi-\-  x~„  yi  =  aiX;i  +  y3, 
x^  =  b^y^  -f-  x^ ; 

prove  that 

^4  =  -  2/1 


-b, 

-1 

0 

0 

0 

0 

0 

1 

—  rt, 

-1 

0 

0 

0 

0 

0 

1 

-^3 

1 

0 

0 

0 

0 

0 

1 

—  ^2 

-1 

0 

0 

0 

0 

0 

1 

-b. 

-1 

0 

0 

0 

0 

0 

1 

-a, 

-1 

0 

0 

0 

0 

0 

1 

-b. 

12.    Given  x  =  bi/(ai-\-y),  y  =  b.,/(a.,-{-z),  z  =  b.,/{a^-{-u), 
and  w  =  ^4/^4;  prove  that 


x  =  bi 


«,   -1    0 

bs     «3     —  1 
0      b^     a* 


«!   -1    0       0 
bo     tto    —  1    0 
-1 
0^4 


'2         <^2 

0      ^3      as 
0      0^. 


(Take  for  variables  x,  xy,  xyz,  xyzu,  and  eliminate  the 
last  three.) 


Solve 

13.    ax  -\-by  - 

-  cz  —  2ab, 

by  -\-  cz  — 

-  ax=  2  be, 

cz  -\-  ax— 

-  by  —  2ac. 

20. 


14.  (c-{-a)x—(c—a)y  —  2bc, 
(a  -\~b)y  —  (a  —  b)z  =2  ac, 
{b -\-  c) z  —  (b  —  c) X  =^2 ab. 


DETERMINANTS. 


3G1 


-ay-V  '^2  +  ^' 
=  ax  -{-  dz  +  c, 
—  hx-\-dy^-f' 


0 
0 
0 

1      0 


0 
0 
0 
0 


1    -61 


)    z-hjia^^u), 


0  0 

-1  0 

a3  —  ^ 

?)4  «4 


and  eliminate  tbe 


15.  {z-\-x)a~{z~  x)b  =  2yz, 
{x-\- y)b  -  {x- y)c  =  2xz, 
iy^-z)c  -{y-  z)a=2xy, 

16.  X -{- ay -{•d^z-\-a^u-\-a^ —  0, 
x-\-by-\-  hh  +  hhi  +  ^*  -=  0, 
X  -{-  cy  -\-  c^z  -\-  c\l  +  c*  =  0, 
a:  +  dy^d'z^  dH-\-d'^  0. 

17 .  X  ^  ay  -\-  ah  +  a"''w  =  c?, 
^'  +  ^y  +  ^'^z  -[-  Z>'w  =  a, 
^  +  <?y  +  ^^^2;  +  c^w  =  h^ 
^-\'  dy-\-  d'^z  +  <^"^i^  =  c  ; 

and  if  a,  ^,  c,  cZ,  are  the  roots  of  the  quartic 
determine  x,  y,  2,  w,  in  terms  o( 2^\>  2^^^  P^^  P*- 

18.     w-f-    ^-\-  y-\-    z  —  h, 

au-\-  bx-]~  cy  -{-  dz  —  /, 
d^u  +  b'^x  -\-  &y  ■\-  d'^z  =  m, 
aht  +  b^x  -{-  c^y  -{-  dh  ~  n. 

19.  Show  that  either  of  the  following  systems  of  equations 

can  be  reduced  to  the  other  : 

(1)     x\  +     :r,  +    0.-3  =  ?^„  (2)  ?/i  4-  ay,  +  ahj.^  =  Vi, 

axi  +  bx2  +  cx-A  =  U.2,  2/1  +  %,  +  ^'/A  =  v,, 

a^x\  +  ^'0:2  +  c%  =  W3 ;  2/1  +  ^3/2  +  ^Va  =  '^'3- 

Generalize. 

20.  There  is  a  certain  rational  integral  expression  whose 

value  depends  on  that  of  x,  and  into  which  x  enters 
in  no  degree  higher  than  the  third.  Its  value  is  4 
when  a:=0,  is  9  when  x  =  l,  is  20  when  x=2, 
and  is  49  when  x  =  S.     Find  the  expression. 


3G2 


DETERMINANTS. 


Solve 


21. 


X       ,       y      ,       z       ,       u 


+ 


+ 


+ 


(t  -{-  k      h  -^  k      c  -{■  k      d  -{-  k 


a?       .      ^      I       z       ,       u 


-f 


+ 


^-- 


a-\-  L      b  +  I      c-\-  I      (1+  I 


X       ,       y_     ,       z       ,       u 


-  + 


+  --^-  + 


a  -f-  m     lf-\-ni     c-\-m.     d-{-  m 


a?       ,      _^      ,       2       ,       ZA 


+ 


+ 


+ 


a-\-  71      h  -{■  n      c  -{-  n      d  -\-  n 


1, 

=  1, 
=  1. 


22. ^ [_  (fillL^K^JZl^  =  0 

bz  -\-  C7J  h  —  c 

ex  -{-  az  c  —  a 

^       ^  (^  -  a)  (b  -  c)  ^  Q 
ay  -\-bx  a  —b 


23. 


1 

X 

X 

X 

X 

1 

c 

b 

X 

c 

1 

a 

X 

b 

a 

1 

=  0. 


24. 


25. 


26. 


0  1  1  1 

1  id' -{-by         a'  b' 
1  ~     x'  {x'-^-b'Y        ¥ 

1         x'  a'  {x'-\-o?y 


-0. 


a~b      X  X  X 

X         X         X      d—  a 


=  0. 


a 


3 


=  0. 


{a-^xf       {b-\-xJ       {c.^x)^ 


h 

1. 

1' 

=1, 

— 

=  1, 

VI 

- 

=  1. 

-  n 

), 

X 

0. 

r=0. 


'4 

'.4 


=  0. 


DETERMINANTS. 


363 


=  0. 


+2f 


27. 


Determine  a,   b,   and  c  so  that    the    two  systenis  of 
equations 

cw  +  bi/  -  cz  =  /,  a,.r  -f  /?,?/  4-  y\Z  =  ^i, 

aa:  ~  irt/  -f-  C2  -  ?7i,  aa.T  -f  ^S^y  -f-  y-i^  =  '^w,, 

—  «^-  4-  Z^y  -f-  cz  =  7i ;  a;,x-  +  iSa^Z  +  y«2  =  '^i  \ 

may  be  satisfied  by  the  same  values  of  x,  y,  z. 


Apply  to  the  case 

ax  +  bi/  —  cz  —   4, 

ax  —  by  -f-  cz  =    8, 

—  ax  -\-  by  -\~  cz=  16, 


2a; 


2/+32 


3a;  +  2y-2z---- 
—  a;+    2/+    2  = 


9, 

1, 

4. 


28.  Solve    2aT  +  3y-42_3a:  +  4y  -22_4a;  +  27/-32 

a;  4- 5  5a;  4a;  — 1 

6 

29.  Eliminate  x,  y,  and  z  from 

(«,.l'  +  %  +  ^>,z  -}-  Ci)  /  w  =  (^3^-  +  ct'zy  +  ^i2  +  c.i)/v 
-~  {b.,x  +  ^i3/  +  a^z  -f  C3)  /■m; 
=  1  —  CiX  ~  c^y     c^z  ; 

icx-\-vy  -{-luz  =  1. 


30.    Determine  a,  given 
^  +  y  +  2  +  ^  =  0, 

a      0      c       e 


ax  -^^  hy  -\-  cz  -\-  dw  '=  Oy 


a 


e^ 


31.    Solve   /i  (JiX  +  '^»i3/  +  ^12)  =  am^  -\-  bn^, 
k  U'lX  +  77i.^y  +  n.^2)  ^  am^  +  67^2^ 


^ 

.^^> 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


k 


A 


//       ^^^."^ 


4s> 


LO 

II  1.1 

11.25 


»    1^    |2.0 
MSB 


u 

muu 


6" 


^J" 
■> 


Photographic 

Sciences 
Corporation 


23  WEST  MAIN  STREET 

WEBSTER,  N.Y.  14980 

(716)  872-4503 


^^^ 


364 


DETERMINANTS. 


M 

I 


32.  If  Xi,  y, ;  r^j,  ?/2 ;  ^3,  3/3  are  the  values  of  x  and  y  that 

satisfy  each  possible  pair  of  the  equations 

a^  +  h.^  +  <?3  =  0, 
^33^  +  hy  +  ^3  =  0, 
prove  that 

1 1  .'^-2^3 1  =  I  aiV3  r-^  !l  aA  1 1  «'A  1 1  «3^i  I}. 

33.  The  equations 

x  +  3i/-\-6z-\-Su  =  S4:,  x+2i/-\-6z-{-4ic  =  S6, 
x-i-  y  +  22!+  w  =  13,  a;4-3y  +  82  +  5w  =  51, 
have  for  sole  solution  x=l,  y-—2,  z  =  S,  w  —  4,  but 
on  attempting  to  find  the  value  of  u  by  indetermi- 
nate multipliers,  on  adding  together  the  equations 
multiplied  respectively  by  1,  a,  /?,  y,  and  equating 
to  zero  the  coefficients  of  x,  y,  and  z  in  the  resulting 
equation,  we  obtain  the  incompatible  equations, 

1+    a+     p-\-     y  =  0, 
3+    a  +  2y8  +  3y  =  0, 

6  +  2a  +  50H-8y  =  O. 

Explain  the  paradox. 

34.  Eliminate  x,  y,  and  z  from 

ax  -{-  by  -{-  cz  —  1  =  bix  -{-  a^y  —  z-\-c 

=  (?ia7  —  y + aiZ  +  6  =  —  a;  4- Ci3/ +  ^i2  +  <3J  =  0. 

36.    Eliminate  w,  v,  w^  a:,  y,  z  from 

axU  +  h^v  -f  CxW  =  0,  aia;  -f-  )8,y  +  yi2  =  w, 
a^u  +  ^2^  +  Cat/;  =  0,  a^x  +  fty  +  722  =  v, 
o-jW  +  h.^v  +  Cat^;  =--  0  ;       a.^x-\r  fty  +  732  =  ^^; 

and  prove  thereby  that 

I  a\hiC'i  I  X  I  ttifty,  | 

=  I  a  ,ai-f -  i  ittaH-  Citta ,  a2^,+  ^2^+  C^ft ,  ^aV  1+  b^-^i^  Ciy-i  \ . 


X  and  y  that 
)n8 


52-f  4tt  =  36, 

8z  +  5w  =  51, 

=  3,  u  =  4,  but 
6  by  indetermi- 
•  the  equations 
f,  and  equating 
in  the  resulting 
I  equations, 


+  a  =  0. 


DETERMINANTS. 


365 


36.  Eliminate  w,  v,  and  w  from 

ai^  -j-  ^v  +  /7^^  =  ^w» 

^M  4-/y  +  <?^^  =  ^^ ; 

and  w,  V,  ?f,  a:,  y,  2;  from  the  three  preceding  equa- 
tions combined  with  the  three  following : 

ax  +  hy  -\-  gz  =  u  —  \x, 
hx-\'by  ■\-fz  -=v  —\y, 
9^ -\-fy  ■\-cz^w-kz\ 

and  reduce  the  two  resultants  to  the  same/o?'m. 

37.  Eliminate  first,  x,  y,  z,  second,/,  g,  h,  from 

aw  -\-  hy  —  gz  =0, 
hio  -\-fz  —hx  =  Oy 
cw  -\-gx-fy  =0. 

38.  Show  that 

o>\-\-  ^\-\-  Cx       a^a^  +  i,Z>2  -f  CiCa 

^laAP  +  l^-i^ur  +  kia^r. 

39.  Prove  that 

a^a^ + 6361 + c^c^ + (Zgfii ,  ajaj + 6363 + ^3^2 + d^d^,      a^ + ^3"  +  c^  f  d^ 
Generalize. 

40.  Let  Ai  =  |aiV3|,  ^2=laifty3|,  and  t!!kxt^^=\AxB,iC^\ 


then  will 


0  oo  /3o  yo 

nto  ^i  ^2  ^3 

^>o  /A  ^2  i?3 
^0 


Cj     Gj     C3 


^,y  1+^372+^373 


=  |ao^ie2|  |a„^,y2|  +  |«o^i^3|  |aoA73|+l«o^2^3|  lOoftysl- 


3G6 


DETERMINANTS. 


State  this  ilieorem  in  the  cases 


1°. 

2°. 


A,  =  A.^,  oo  -^  ao,  Po  =■-  bo,  yo  =  Co ; 

A,  L^  Aa,  Oo  =  Co,  /3o  -~-  bo  =  yo  =  Co  —  0. 


(1) 


(2) 


41.  Given  w,  =  a,a: -f  ^ly  +  ^i2  +  ^i  =  0, 
Mjj  =  a^x  -\-  h^y  +  CjZ  +  r?,  =  0, 
Ws  =  a^x  4"  ^32/  +  C32  -f  ■  ^3  =  0, 
W4  =  a^a;  +  ^4^  +  CiZ  +  c?4  =  0, 
W6  =  a^x  +  ^53/  +  C52;  4-  c?5  --=  0.  ^ 

1**.  Determine  the  value  of  x  that  will  satisfy 
ttiWi  +  agWj  +  a3W3  +  aiUi  -\-  a^Ur,  —  0, 

biUx  -\-  b.iU.i  +  ^3^3  +  ^4^4  +  65W5  =  0, 

CjWi  +  C2U2  +  C3W3  +  C^U^  +  C5W5  =  0.  J 

2°.  Eliminate  z  from  the  group  of  equations  (1),  taken 
two  by  two  in  every  possible  way  ;  from  the  result- 
ing ten  equations  form  two  equations  by  a  method 
similar  to  that  by  which  the  set  (2)  was  formed  from 
the  group  (1) ;  and  determine  the  value  of  x  that 
will  satisfy  these  two  equations.  Show  that  this 
value  of  X  is  the  same  as  the  value  obtained  bv  the 
solution  of  the  set  (2). 

Apply  the  preceding  to  the  equations 


X—    y  -\-^z- 

-3-0, 

Zx-{-2y-bz- 

-   5-0, 

4a;+    y-\-4iZ- 

-21-0, 

x-\-'6y-\-^z- 

-14-0. 

Generalize 

42.   Eliminate  x,  y,  z,  and  h  from 

o-xX  +  /?,?/  +  ViZ  =  0,     02^:  +  fty  -I-  yaZ  =  0. 

ci\X  +  hy  -f-  b-jZ  ^  b^x  -f  a-^y  -f  b^z  ^  h,x  -f  b^y  +  a^z 


(^0  > 

=  Co-0. 


DETERMINANTS. 


3G7 


(1) 


U  satisfy 
..==0,1  (2) 

luations  (1),  taken 
;  from  the  result- 
eitions  by  a  method 
2)  was  formed  from 
te  value  of  x  that 
Show  that  this 
e  obtained  bv  the 


43.  Ehminate  x,  y,  z,  u,  ki,  and  k^  from 

<^ix  -f  i8,y  -f-  y,2  +  B,u  =  0, 
O'iX -i- p.^  +  y.,z -]- 8.,u  =  0, 

ax  +  Ay  +  f/z  +  ^M  __  hx  -\-  by  -{-fz  +  tnu 

—  9^  ~^fy  -\-  cz^nu  _  Ix  +  ^y  -\-nz-\-  du 
Vi  +  72^1  +  Vs^'a  81  +  SjZ;!  +  8»^2 

44.  If  «i^  +  h^y  +  gi^  ^  g.^  +  &2?/  +  c^z  ^  ftaa^-  +  ^3y  +  ga^ 


w 


!<; 


then 


X 


y_ 


AiU  +  ^2^  +  -4si^     -5iW  +  -^2^;  4-  A"^ 


» 


in  which  A^,  A^,  etc.,  are  the  inverse  elements  of  ai, 
aj,  etc.,  with  respect  to  |  ai^^Ca  |. 

45.  Eliminate  x,  y,  and  z  from 

«"  +3/'  +z'  =^, 

(ar  -  of  -{-y'  -\- z'  =  (k  -  d)\ 

{x~a,y-\-{y~h,y  +  z'  ^(k-dO\ 

{X  -  a,y  +  (y  -  ^^2)^  4-  (2  -  ^2)'  =  (^  -  c?2)^ 

46.  Apply  the  results  of  Examples  1  and  2,  pages  348  and 

349,  to  prove  that  if  the  value  of  a  determinant  be 
zero,  the  determinant  may  be  transformed  into  an- 
other of  the  same  order  in  which  all  the  elements  of 
a  row  or  of  a  column  shall  be  zero. 


Transform 

2  3        4     3 

3  4        1-2 
6       5-2-5 

y,Z  =  0.,                     1 

10       12     3       4 

M4-^».V  +  ^»?.  1 

into  a  determinant  of  the  fourth  order  with  a  col 

yi  4-  72^'        1 

umn  of  ze 

ros. 

3G8 


DETERMINANTS. 


Similarly  transform 


13        2-4 
4-2-13      5 
2     8        7-11 
3-1-9       3 

Apply  the  above  to  prove  the  rule  for  the  multiplica- 
tion of  determinants. 

47.  Eliminate  a,  h,  c,  d,  e,ffrom 

ax^  +  2bx7/  +cy^  ■\-2ex-\-2fy-\-d^0, 

ax -{-    h(xp+y)  -{- cyp  +    e  +  fp      =0, 

a    +    b(2p-j-xq)-\-c(p^+yq)  -f  fq 

b{Sq-\-xr)i-c{Spq-{-y7')  -{-  fr 

b(4:r+xs) -}-c(4:pr  +  Sq^-\-ys)  +  fs 

b{5s  -\-xl)  +  c{bps  +  10qr-\-yt)-{-  ft 

and  evaluate  the  resulting  determinant. 

[Omit  the  first  three  equations,  —  this  eliminates  a,  e, 
and  d;  in  the  remaining  three,  take  for  variables 
b  -j-  cp,  cq,  and  bx  -f  cy  +/.] 

48.  If-^    +£zi-Cy  +  Du  =  0, 

Az-B  +Cx  +  Dv  =0, 
Ay  +  Bx-C  +I)w  =  0, 
Au+Bv  +  Cw-B    =0, 


=  0, 
=  0. 

=-0, 


then  will 


A' 


B" 


1  —  x^  -  -  v^  ~w'^  -\-2  xvw      l  —  u^—y'^—w^-{-2  uyw 


Q2 


U" 


l-~u'^  —  v'^—z^-{-2uvz     1  —  xi^  —  y^  —  z^  i-  2xyz 


49.    Eliminate  u,  v,  and  w  from 
uxi  +  vyi  -f  ivzi  —■  0, 


ux, 


a 


4-  vy2  +  WZ2  =  0, 


h 


h    b    f 
9    f 


u     V 


g    u 

V 
C       IV 

IV    0 


0. 


DETERMINANTS. 


3r,o 


be  multipUca- 


+  h 

)  +  /s 
mt. 


=  0, 
=  0, 
-0, 
-0. 


eliminates  a,  e, 
le  for  variables 


50.    Show  that  the  system  of  equations 

a  {h  —  c)      b(c  —  a)  .  c{a  —  h)  __  ^ 
a  —  a  o  ~  p  c      y 

«(ff-Y),  ff(Y-a)  I  y(«    -fi)  ^.q^      . 
a~  a  fi  —  b  y  —  c 

is  satisfied  by  either 

a  —  b'\-P  —  a  —  b  —  c-\-y~P~o  —  a-^a 

or      aPy  =  aby  ■■=  aftc. 

Ki      Tf    ^^  "h  Q.V  —  rz  _  ^a:  —  ^y  4-  <^'i2:  _  —  <^<-^'  +  ^7/  +  ^^^ 
a'H^'      ~       «'  +  c''       ~  b'  +  (^ 


then  will 


.1- 

y 

z 

c 

a 

b 

b 

c 

a 

=  0. 


52.    Eliminate  x,  ?/,  and  z  from 

a{x-l)  +  b(i/-l)-}-c(2-l)  =  0, 

=  0. 


a; 

y 

z 

c 

a 

b 

b 

c 

a 

[Reduce  the  determinant  to  the  form 


53.    If 


ax-\-by-{-cz  x-\-y  -\-  z 

ab  -\-bc  -\-ca  a-\-b  -{- 

Ix  -f  'my  -{-nz  =0, 

ax  —  by—cz  =u, 

—  ax  +  %  ~cz  =v, 

—  ax  —  by  -\-  cz  =  w, 
aux  +  bvy  -f-  cwz  =  1, 


-0.] 

I    m    n 

X   y     z 

U    V       w 

=  0. 


then  will 


vi^  -\-  n^ 


n' 


+  /•' 


P  -\-  m' 


{bn  -j-  cinf      (cl  -f  any      (am  -f-  ^0'' 
^x'  +  y'  +  z\ 
and  4  a''b'c\x'-\-y'+zy-(a'i-b^i-c'){x'-\-y'^z')~\-l=0. 


870 


DKTERiMINANTS. 


i'i- 


64.    Exhibit  in  a  single  equation  the  result  of  eliminating 
u,  X,  y,  z  from 

ax  -\-  hy  +  gz  ~  ciiU  -f-  A-i^?, 
hx  -f  by  Arfz  =--  h^u  -f  Aiy, 
d^  -^Jy  -hcz  =  ciu  +  Xiz, 
ctix-i-  b^y-\-  Ci2  =  0  ; 

and  V,  Xi,  y,,  Zi  from 

ax^  +hyi  -\-fjZi  -=  a,t;  +  X.,-^, 
hxi  -j-hyi  -\-fzi  -^'biV-i-X^y, 

0^\  +  fy\  +  <^-i  =  ^1^  +  Kz, 
(iiXi  4-  5,y,  +  CiZi  =  0. 

55.    If  ai-\-biy-\~CiU  =  a7V-\-b2-}-C2'/'  =  a3X-\-ba,w-\-Ci=s, 

ajid  wa;  =^  vy  —  wz  =  1, 
then  will 

«3  ^3       ^3  —  5 


56.    Eliminate  X  frori 


x^  '/ 


a  —  X     ^>  —  X     c  —  k 


=c, 


a:a;' 


+ 


i^ 


zz 


a  - \     b- 
57.    Eliminate  X  from 


+  -^-—  ..  0. 


X      c-X 


X' 


+ 


w' 


a  —  X     6  —  X     c  —  X     ^  —  X 


=  0. 


XX' 


a  —  X     6  —  X     c— X     y  —  k 

58.    Eliminate  ?^  t?,  and  w  from 

joy  w  +  g-y  V  -\-  r^/w~0, 

f^  I  u  +  '^^y  ^  +  ^y  w^O, 

F(q  —  r)u-\-  Q(r—p)v-\-  R{p  —  q)w  —  0. 


63 


DETERMINANTS. 


371 


f  eliminating 


-{-biW  +  c^—Sf 


0, 
0. 


—  q)w=0. 


59.  If  (a,  -  a,)^  +  (ft      /8,)' =  r,' +  >•2^ 

(a,  aa^  +  (A  -  ft)^  -  nM- r,', 
(a.  -  a,)'  +  {Px  -  P.y  -  n'  f  ^V, 
(a,  -  a.,)'  +  (ft  -  ft)'  =  r,'  +  r,', 

(a,-a,)'4-(ft~ft)'  =  r,»-M'A 
(aa  -  aO'  +  (ft  -  ft)'  =  r.,'  -f-  u\ 

then  will  r,  '  ~\-  r./  '  +  ?3-'  +  u  '  =  0, 

and  ^,ri  '  +  ^i,r,  '  +  ^U^a" '  +  ^4^*"'  =  0, 

in  \^'hicli  A  is  either  a  or  ft 

60.  Eliminate  u,  x,  y,  z  from 

u-{-x  +  y-{-z  =  0, 
au  +  ^•^'  +  <7/  +  c?z  =  0, 

a/2C-{-b/x-\-c/7/-\-  d  I  z  =  0. 

61.  Eliminate  w,  v,  ?<;,  .r,  ?/,  2  from 

Za;  +  wy  +  wz  =0, 
^w  +  lYlV  +  wi^'  =  0, 
fyz  ^  (jxz -\- hxy  =  0, 
ux     _     vy     _     i<^z 


62.    Eliminate  x,  y,  z  from 


a' 


a 


w. 


x  +  y-\-z=--/ui-/v-\-/ 
ax-^by+cz  =  /{ax)  +  /{hj)  +  /(cz), 

X 


u 


0+V(yA)+V(V^)  =  o. 
■^^)+y/(i-^«)+2i 

and  reduce  the  resultant  to  the  form 


/(I  -  be)  +  y/(l  -  ca)  +  z/(l  -  ab)  =  0, 


^M  +  ^V  +  C'lO  =  0. 

63.    Eliminate  ar  and  y  from 

(1  +  x)/(a  —  y)  =  /a  +  x/a, 
{l  +  x)/(b-y)=:/b  +  x/p, 
{l  +  x)/{c-y)  =  /ci-x/y. 


37'J 


HETKTlMlNANT.'-' 


64.  Kliininiile  r,  y,  z,  s  from 

{s  -  y)(s-z)^ayz, 

(a-.r)(6«     ?/)      cxj/, 

65.  Given         .rys  -  a  -f  t/s  (y  +  2)  -'  ^^  1"  -^  (-  ^^ ''') 

show  tluii  4  Cvyzf  -  {ah  +  ^^c  +  ca)  (xj/z)  +  «^c;  -  0. 

66.  Determine  \,  /tx,  v,  given 


a-a   b  —  P   c-y 

I  m  n 


=  0. 


Ik  +  7>i/w.  +  ni/  =  0, 

X'  +  ,i,'^  +  v'^-l. 

67.  Find  ^^,  v,  w,  given 

/w  +  ?uv  +  mo  —  i\ 

68.  If  \{ax^-hy-V<JZ-Vlw)■\-l^(hx■\■hy^-fz-\r'tnw) 

+  V  {(jx  -\-Jy  \-cz\  n w)  =  0, 
a\  +  hfi  +  (jv  =  itX, 

yX  -\-ffj^  H-  c'v  =  uv, 

69.  Resolve  a  system  of  three  equations  in  three  unknowns 

of  which  one  equation  is  quadratic   and  two  are 
linear. 

70.  Apply  the  method  of  Example  9,  p.  355.  to  resolve 

x'  +  y'-{-z'--^h 
aiX-\-biy  +  CiZ=^ux, 


DETERMINANTS. 


373 


X  (2  -f  ^0 
z)  +  abc  -=  0. 


iin  three  unknowns 
atic   and  two  are 


355,  to  resolve 


and  c'oinparo  the  viiluos  of  u  given  by  iho  ri'sohition 
and  tlial  obtaiiKMl  by  oliniiniiting  x,  i/,  and  z  IVoni 
tlic  last  tlnce  equations. 
Genoralizo. 

71.  Mliniinaio  ?<,,  i',,  ?/,.^,  I'j  from 

(f/f/      Z>r7)  ^'1?^,  —  X2  t/i  -    a;,?/,  4-  «i^'i  —  WjVj. 

72.  Eliminate  x,  y,  and  2  from 

a;-|  y-h2---=0, 

^fj;'^  4-  /;y'  +  cz'  +  2/?/z  -f  2fjxz  +  2  Aary  -  0, 
rti-r*  -}-  /;,,?/  +  C;iZ^  +  3  a.^a;^/  -f  ^o^^xh  +  3  />,.r?/' 
+  3  b,fz  +  3  ^ia:2''  f  3  c,y2'  +  6  dxijz  --  0. 

73.  EHminate  x,  y,  and  z  from 

^.2_|_y  —  2^73:?/ —  0, 
y»  +  2'-2%2-0, 

and  assuming  the  resulting  relation  to  hold  among 
ff,  h,  and  k,  find  the  H.C.F.  of  the  functions 

u^  +  v^  —  2  guv  —  (1  —  ,7"), 
v"  -\-w'-2hvw  -l\~h'), 
w'^-\-u^  —  'J^  kwu  -  (1  —  ¥). 

74.  Eliminate  x,  y,  and  z  from 

{f\  —  u)yz-]-g^zx^-h^xy  =  0, 

fiy^  +  (^2  --  y)  zx  -h  /«2a;y  =0, 
fzyz-\-gzZX-\-{h-u)xy  =0. 


374 


DKTERMIN'ANTS. 


76.    Eliininato  x,  y,  and  z  from 

••i^"'  -\-  Kj/'^  h  c^i^'  I  -  '^J'lj/z  +  li  (//,!     u)  zx  -f-  2  /^;.r//       ( ), 


?•.    Expand 


X'  -  <r 


y' 


x' 


?/' 


y 


0. 


77.    Expand 


a'  -1/ 


z'-C 


x'-  z' 


—  X*  —  ?/' 


-0. 


and  reduce  the  expanded  equation  to  the  form 


a  ,-3 


ax 


a  —  8' 


+ 


>v 


•^-•i 


^»*'- 


+ 


0. 


c'  -  s' 


s'  =  .r'  -f  y  +  2 


78.    If  a,  ;8,  y  bo  the  values  of  u  satisfyin 


rr 


X  ~{-  a~u 


V 


X 
X 


y 


u 


z-\-  C  —  %l 


0. 


and  if 


a-y  —  z 

X 
X 


y 

z  —  X 

y 


C  —  X 


y 


0. 


then  will 


(o--a)(cr-/8)(cr-y)==0, 
in  which 

cr3a  +  )8-fy  — a  — Z>— c. 

79.   Solve  r»  +  2a;V  +  2a;y  (y  -  2)  +  y'  -  4  =  0, 
a;' +  2a:y  +  2?/'' -  5y  +  2  =  0. 


DETERMINANTS. 


375 


0. 
0. 


-0. 

to  tlie  iorin 


?(- 


-0, 


-0, 


-4  =  0, 


80.    fciolve  ar'-f  3Ay-(u7/   ^  +  /  -  0, 


3i;V      3.1:^1^      3/-  y}- 3-0. 


81.    Solvu 


(^-  f-i)(y-l-i)    G. 

[Translbrm  by  /^      .r-|-  ?/,  v  -  .ry,  and  (sliininatc  ?/.  | 


82.    Solvo 


X -\-  xy  \  J/ -5. 
[Traiisfoini  by  a  ^X'\-i/  -\-  xf/,  v  —  (x -f-  //) x>/]. 

83.  Solve  x'-l-y'-i-z'      (a^\-u)\ 

(a,  -  ;r)' +  (/>,      !/)' +  z' ^  (y  \  u)\ 

(a,  -  xf  -h  {i>.  -  y)'^  -f-  (^•.   ^y^  -  (8 + uy. 

84.  If    a  =  a,A  f-  rt.j/u,  f  «:,i/,     )8=  />,A.   f-  /^a/x   f-  ^a*', 

a:r,^  -f  /?.r,'^  -}-  y.i'.  +  8  -  0, 
cur,;'  +  i8.r/  +  yx,  +  8  =  0, 
a:r:,3  f  /S^a'  +  y^;,  +  8  -  0, 
in  which  .r,,  .r.j,  and  arj  are  the  roots  of 


a;' 


-f-/Af''  +  !?^  +  ^'  =  0, 


show  that 


1 


7^       <Z 


<^3 


-0. 


What  does  this  equation  become  'i(  Xi  =  x.t? 
What  does  it  become  if  Xi  =  X2  ~  x^? 

[85.    If   ^1^1^  +  ^i-'^t'  +  ^1^1  +  ^1  -.  ^•^■^t^  +  ^•^•^'i'  +  ^-^^1  +  ^^^ 


DETERMINANTS. 


m 


show  that  Xi  and  x.i  are  the  roots  of  the  quadratic 

=  0, 


x" 

—  X 

1 

A 

B 

c 

B 

C 

D 

in  which  A,  B,  C,  and  T)  are  the  four  determinants 
of  the  third  order  that  can  be  made  from  the  array 

'«!   hx  Cx   dx 
'  a^  h.i  C;j  o?2 

^  ^3     ^3    ^Z    ^3- 

86.    If  Ux^^  <  =  w/  +  V2'  ----- 1,  then  both 

\a\x-  Uxf  +  ^'  (3/  —  f^iT  \  (^^2  +  yv^  —  ly 
=  {a''(x~u.,y  +  b''(i/-v^y\(x2ix+yvx-iy 

and  [  a'^(x  —  Uxy  +  i^y  ~  '^lY ! !  (^^2 — y ^2)'  —  (a;  —  Wj)^ 
=  [a'(x—u^y+b%i/-Viyi  {(xvi~yu,y 

are  satisfied  by 

X      V      1    = 


y 

Ux       Vx        1 
U^      Vo.       1 


0. 


87.    If  rxhA(hx'  +  ^')  =  r,hAW+P)  =  rAh2(V+P), 


then  will 


hi     hji^    r^n 
hi     hhx     nrx 

h       hyhi       TxTi 


0. 


88.    If^  +  .^=^>^ 


a 


a 


+ 


y/  - 


^^1    L.V.Vl 


CCu/o 


a* 


28, 


a' 


28, 


n         I  7  'i         I      /»  C\i 


a 


28' 


DETERMINANTS. 


377 


ihe  quadratic 


,ur  determinants 
.  from  the  array 


XVi 


-2/^0' 


=  r3W2(W+^'). 


l+o.. 


show  that 


1     .V     y 
1     .Tj     y, 


ah 


uuia. 


^  OOiO] 


0 


89.    Given  that  aj.r'^f- 2 />,a,--f<",  =0  and  a.j^'*  +  2 ^^a: -|- r./ 
have  a  common  root,  dttermine  it. 

Apply  to  case  of 

(990:r^- 441  a;   -5390  =  0. 
(  825:r'- 428  ;f    -4620  =  0. 


90.  Given  that  oa;"''  |- 3  i.r  +  3 (?:c -{- cZ  has  a  square  factor, 

find  it. 

Apply  to        2940r' 4- 812  a;'- 8385  a; -G300. 

91 .  De lermine  the  condition  that  ar* -f  3  hx^  +  3  r.r  +  d  --  0 

and  ax^  +  2/3a;  +  y  =  0  shall  have  a  common  root, 

and  find  it. 

> 

Apply  to         I    30. r'  +  x'  -f  35 a;  -f  204  -  0, 
U10.r'^- 23a; -357  =  0. 

92.  Determine  the  condition  that 

ax^  -f-  4  ^ar*  +  6ca;^  +  46?a;  +  e 
and  aa;'  +  3/?.r'  +  3ya;  +  8 

shall  have  a  common  linear  factor,  and  find   the 
common  factor. 

93.  Determine  the  condition  that 

ax^ -\-  ^hj? -\-  ^cx^  -\-  ^.dx  -{-  e 
and  ax"-  +  4^a;'+  ^jx\  +  48a;  +  « 

may  have  a  common  quadratic  factor,  and  find  the 
common  factor.  ' 

Apply  to  r60.r*-4a;'  +  37a;'-      a; +  28, 

I  90a;*  +  3 a.-=»  +  84 x^  +  22a;  +  50. 


378 


DETERMINANTS. 


94.  Determine  the  conditions  that 

UyX^  -{-  2  h^x  -\-  Cx  --  0, 

a,x'  +  2b,x  f-c'  =  0, 
shall  have  a  common  root. 

95.  Determine  the  condition  that 

ax*  +  4  hx^  -\-  (j  cx"^  -{-  ^  dx -{■  €  ~  0 
shall  have  two  equal  roots. 

36.    Determine  the  conditions  that 

ax*  +  Ux""  i-Qcx''  +  idx-{-e  =  0 
may  have  three  equal  roots. 

97.  Determine  the  conditions  that 

ax'  +  5hx* -\-  lOrx' i-  lOdx'  +  5ex -i- g  =^0 
may  have  three  equal  roots. 

98.  Determine  the  conditions  that  ax -{-hi/  may  be  a  com- 

mon factor  of 

and  a-^x^  -|-  3  h.^x^y  -\-  3  c.^xy"^  -f  d.^^. 

99.  Determine  the  remainders  in  the  process  for  finding 

the  H.  C.  F.  of/(a:)'"  and  F(x)''+\ 

100.    If  x"^  +  h.x""-'  4-  b.,x"'- '  + be  divided  by 

a:"  +  a,a;'*~' +  «2.^'"~■^  + , 

then  will  the  coefficient  of  the  rth  term  of  the  quo- 
tient be 


(-1) 


r    1 


1 

1 

0 

0 

i. 

a, 

1 

0 

b. 

a., 

«i 

1 

b. 

«3 

a, 

a, 

br-l      (-h    1     (^r    -1     Or    ^ 


DETERMINANTS. 


379 


=  0 


=  0 


|-Z>2/maybe  a  corn- 


process  for  finding 


ivided  by 


^tli  term  of  the  quo- 


Ir-  3 


101.  For  what  value  of  a  will 

(a  -2)^:'- 2a:  +  5a -2 
and  ax^  ~  5  .t'  +  4  « 

have  a  common  factor? 

102.  For  what  values  of  y  will 

x^  —  4  x'^y  -\-  xif  -  -  (?/  —  1)* 

have  a  common  factor  ? 

103.  Find  the  relations  that  must  hold  among  r/,  h,  and  c, 

that 

a:i^  -\-hx-\-  c 

2,\i{ia{\--c)x^-\-h\{\  —  c)-\-ac\x-\-c 
mav  have  a  common  factor. 

104.  If  2.r*  —  .t'^  +  «  =  0  and  x^  —  x  -\-  b  =  0  have  a  com- 

mon root,  then  must 

{4:b-  1)  (2P  -  «)  +  («  +  2i-^  -  /;)'^  =  0  ; 

and  if   &  +  2  =  0,    find   the  values  of  a  and  the 
resulting  equations. 

105 .  If  Oix'^  -\-2biX-\-Ci  =  0  and  a^x"^  +  2  ^.^a:  -f  ^.^  =  0  have 

a  common  root, 

{aiCi—bi^)x^'{-{aiC.i-i-Cia2~2bib.i)x-\-(a2C.j^--b.i^)~0 
will  have  equal  roots, 

106.  If  ttix"^  +  2 bix  +  <?!  =  0  and  a^a;'  +  2  //^o;  +  e^  =  0 

have  a  common  root,  their  other  roots  are  given  by 
ttia-i  I  b^Ci  \x^  -{-\  a,C2  \^x  -\-  C1C2 1  a\bi  \  =  0. 

107.  Eliminate  Xi,  x.2,  X3  from 

(^•2\^2  "T"  '^'3/  ^~  ■^^•2)         a2^2'^3  ^^^^  ^2- 

Form  the  quadratic  whose  roots  are  .r,,  .^-.^ ;  and  the 
cubic  whose  roots  are  .?',,  x^,  X3. 


380 


DETERMINANTS. 


108.    Find  the  condition  that  must  be  fulfilled  in  order  that 


«,     L  T^    •" 


JfL- 


u-\-  a^     u-\- 1) 


u  -\-  r 


shall  have  equal  roots  in  u. 

109.  Find  the  condition  that 

iL  -\-  a''      u  I  f^^      w  -f-  c^ 
may  have  equal  roots  in  u. 

110.  If  ax'  +  4  hx""  4-  6  r'o;''  +  4  t/.r  +  c  =  0   have   a   double 

root,  it  will  be  given  by 

'6{ax^  +  2hx-^cf  =  L 
[For  the  values  of  /and  ,/,  see  (28)  and  (29),  p.  305.] 

111.  Sliow  lliat  if  (he  quartic 

(ix'-\'U7^-\-^cx^-^^dx-\-e^0 

have  three  equal  roots,  then  will 

ad-hc   _ae-\-2hd~^c\_     S(he-cd) 
2{ac-h')         S{ad-bc)         ae+'lhd-Zc^ 

_2(ce-d'')^ 
be  —  cd 
and  prove  that  these  equations  are  equivalent  to 
/-=./=  0. 

112.  Show  that  the  conditions  that  the  quartic  in  111  shall 

have  three  equal  roots  are  the  same  as  the  condi- 
tions that 

ax"^  -\- 2hx -{-  c  =  0, 
bx'  +2cx  +  d=0, 
ex""  -^2dx+e  =  0, 

shall  have  a  common  root,  and  express   them   as 
determinants.     (See  problems  94  and  96  above.) 


DETERMINANTS. 


381 


ed  in  order  that 


113. 


3  have   a   double 
0  and  (29),  p.  305.] 
=  0 

are  equivalent  to 

quartic  in  111  shall 
same  as  the  condi- 


id  express  them   as 
I94  and  96  above.) 


114. 
115. 
116. 
117. 
118. 
119. 
120. 
121. 
125. 
126. 
127. 
123. 


Find  the  relation  that  must  exist  between  (j  and  h  in 
order  that 

2«  -  10//V  +  12  h*2  -F  b</  =-  0 

may  have  a  pair  of  equal  roots. 

For  whut  values  of  2  will  each  of  the  following  equa- 
tions have  ii  pair  of  equal  roots  ? 

x'  +  (2  -  1)  -t-'  f  (z  -   8)  X  -  G  (2  -  2)  -=  0. 

r*  +  'Zzx'  +  (2'  -  52  -  75):i;  -  5(2^  +  52  -  50)  =  0. 

rH  (32  -  2)0,-^  -  (G2  +  15)a;  -  452  =  0. 

x'  -  2(2  +  2)2'"^  +  (82  -I-  3)a;  -  G2  =  0. 

x"  +  (2  +  G)a;^  -f-  (42  +  U)x  +  3  (2  +  2)  ---  0. 

a;'-3a;4-22 -0.       122.  ar''+322- - (1-2)^-4 2='-=  0. 


X' 


Sx'i-  z'  =  0.       123.  2x^  -  32a;  +  2H  1  =  0. 


129. 


x"  -  320;  +  2-'  =  0.       124.  x'  -  ^x'  -f  4a;  +  2  -  0. 

a;*  +  4 a;' 4- 44a;' -  9Ga,- +  2 -^  0. 

5a;^  -  (1  -  2)a;*S5^  -  4*2(1  -  2)*  =  0. 

a;*  +  ^'  +  a;'  +  2  =  0. 

Find  the  relation  that  must  hold  among  t)"*  coeffi- 
cients that 

a;«+G^»a;'^-j-15ca;*-f-20r7.c'+15ea;'  +  G/a;-f/ 
may  break  up  into  the  cubic  factors 

x^  +  3  a^x"^  -{-2>hiX-\-  g 
and  a;'  +  3  a-jX^  -{-  3  hiX  +  g. 

Show  that  the  discriminant  of 

a  (ar'  4-  2/^)  +  ^  {vx^  -\-  v~^  y')  xy 
+  c  {v^x  +  v'^y)  a;y 

is  a  rational  integral  function  of  a,  b,  c,  and  (v^-{-v~^) 
and  of  the  second  degree  in  the  la!=it  of  these. 


382 


DETERMINANTS. 


130.  Prove  that 

{W  -  ah)  {ax'  +  hif  +  2fy  +  2ffx  +  2  hxy) 
-  aP  -  hg'  H-  2fyh  ' 
can  be  resolved  into  linear  factors,  and  find  them. 

131.  Show  that  if 

a  +  ^»  4-  e  =  0 
and  d'h  -j-  h'^c  +  c'^a  +  2mahc  =  0, 
then  will 

ax^  -\-  hy'  +  ^2^  +  2 (mc  -\-  a)yz-\-^ (ma  +  i)2'z 
+  2(?/ii  +  6')x?/ 
be  the  product  of  linear  factors.     Find  them. 

132.  li  x'^-}-2hxy -{-y"^ —  6x    -7y-\-(j  has  linear  factors, 

find  them. 

133.  For  what  values  of  A  will 

2(x^  +  by' -{- z' -yz- 7 xz-{- 2x7/) 
-\(:J'  +  Sy'  +  2z') 
be  resolvable  into  linear  factors  ? 
Find  the  factors  in  each  case. 

134.  For  what  values  of  A  will 

80^;'^  +  8y'  -  iz'  -  8yz  +  6zx  +  S^xy 

-  X  (;r'^  +  y^  +  s'O 
be  resolvable  into  linear  factors  ? 

Find  the  factors  in  each  case. 

1 35.  If  (liX  -j-  viiy  +  Uizf  +  (4^'  +  m.^y  +  n-^zf 

-^  (l^x  i- m^y -{- n,zy  =  x' i- y^  i- z\ 
show  that 

80:1'"'  +  8?/  -4z'  --  Syz  +  5zx  +  SSxy 
==  a  (/ix  +  ??ii?/  +  71,2)'  +  f3  (kx  +  Way  +  rhzy 
+  y(4^'^-w.8?/-f-7^32)^ 

in  which  a,  j8,  and  y  are  the  values  of  X  found  in 
the  preceding  problem. 


DETERMINANTS. 


383 


gx  +  2  hxy) 
and  find  tliem. 


7/;j  +  2(wa  +  ^)^z 

Find  them, 
has  linear  factors, 


^2xy) 
? 


bzx-\-^^^y 


hzx-v^^^y 

{kx  +  trhy-^r'Thzy 
Llues  of  X  found  in 


136.  Find  the  condition  that 

{x'-\-yz){fj  -  cXl  +  kbc)  +  0/  -f-  -0(^  -  '0(l-f-  ^^^) 
-{-(z'i-xyXa-bXli-ka/j) 

may  break  up  into  linear  factors. 

137.  Find  the  condition  that 

k(aiX^  +  ^^i?/  +  <^i-^  +  2/,?/2  -f-  2(/iXz  -\-  2  /(^^7/) 
+  (a,x'  +  i.y  +  c,z'  +  2/,y2  +  2.r/,2-2  +  2  h,xy) 

shall  be  resolvable  into  linear  factors. 

138.  Determine  k  so  that 

Ax'-~9y''~2z''~Syz-\-2xz-\-Sxy 
-^k(x-Sy+z)(x-\-y~5z) 

may  be  resolvable  into  a  pair  of  linear  factors. 

139.  Find  the  condition  that 

k  (aiX^  -f-  3  biX^  -{-SciX  -\-  o?,) 
+  ^(a.^.r'  +  3  b.^x^  -\-3c2X-\-  d^) 

shall  have  a  square  factor,  and  this  condition  being 
fulfilled,  find  the  square  factor. 

140.  Given 

xya^i-f/b'-z'/c'^O, 

Ix  +  "Tt^y  +  92Z  =  0, 

find  the  condition  that  the  ratios  x:y\z  shall  be 
each  single-valued. 

141.  Given 

/V  +  cjY  +  hh^  -  2ghyz  -  'Ifhxz  -  2fgxy  =  0 
and  Ix  -\-  my  -\-  nz  —  0, 

find  the  condition  that  the  ratios  x  -.y  \z  shall  be 
each  single-valued. 


384 


DETERMINANTS. 


142.  If         cur  +  ft/  +  yz  =  0, 

and  fix  +  (J In  +  hl%  =  0, 

and   if  the  ratios  x  \y  \z  are  each  single-valued, 
then  will 

(»-  +  (,'//S)-  +  (Ay)^  -  0. 

143.  Find  the  condition  that  if 

aoc^  -\-  Inf  +  cz^  -f  2/y2  -y  'Igxz  -f-  2  ^xy  =  0, 


anr 


1 


-0, 


a       h  (f  Qi  z  —y  0 

h  h  f  -z  ^  X  0 

q  J  0  y  —  .r  0  0 

b  -0  y  1  0  0  / 

2       0  -a;  0  1  0  m 

"?/  a;  0  0  0  1  n 

0  0  0  I  7)1  n     l''^m'-\-n' 

the  ratios  x\y  \z  shall  be  each  single-valued. 

144.  Given  aa?  +  iy"^  -f  c^  -f-  2/yz  +  2^a:2  +  2  /i.ry  =  0 

and  Ix  -f-  my  +  nz  —  0, 

find  the  condition  that  the  ratios  x\y  \z  shall  be 
each  single-valued. 

145.  Eliminate  ar,  y,  and  2;  from 

ao:  +  /3y  -f-  yz  ^  0, 

{a^(s-:r)J^;^y(s~y)|i  +  Jy2(s-z)i^  =  0, 
2s==a;  +  3/4-2- 

146.  If         u^ax''-^hf-\-cz''^2fyz-\-2gxz-\-2hxy 

be  resolvable  into  linear  factors,  and  if  w  =  0,  then 
will 


a  h  ax-\-hy-\-gz 

h  b  hx^by-{-fz 

ax-\-hy-\-gz  hx-\-hy-\-fz  0 


=  0. 


DETKRMINANTS. 

5«{) 

147. 

Prove  that  the  roots  of 

h  single-valued, 

a  —  X         h             <i 
h         h    -  X        f 

0         f      ^  -  '^• 

=  0 

are  all  three  real. 

See  hist  part  of  problem  3G  of  this  Exercise.] 

148. 

Show  that  the  roots  of 

0        1-0, 

0 
0 

a  —X        h            q        k 
h          b       X         'f         I 
(J            f        c  —  X    tn 
k             I           m      0 

-0 

I 

lie  between  the  roots  of  the  e 
147. 

quation  of  problem 

n 
single-valued. 

\ixz  +  2  h^y  =  0 


iios 


x\y\z  shall  be 


i  +  {yz(s-z)?^  =  0, 


2gxz-\-'2.hxy 
Is,  and  if  u  =  0,  then 

ax-\-}iy-V9A^^' 
hx+by^fi 
0 


149.  Find  the  condition  that  the  equation  of  problem  148 

shall  have  equal  roots. 

150.  Show  that  the  roots  of 

a  —  aiX    h  —  hxX    g  —  g^x   =  0 
h  —  hiX     b—  bxX    f  —fiX 

g  —  Qy^  f  —  h^    ^  —  ^1^ 
are  real. 

[Reduce  to  a  determinant  of  the  form  of  that  in  prob- 
lem 147.] 

151.  Reduce 

ax  +  «!    inx  +  ^1      lx-\-  Ix 

hx  {-hi      bx-\-  bi      kx  +  k^ 

9^+9i     /^+/i       cx-\-Ci 
to  a  determinant  of  the  third  order  with  x  in  the 
principal  diagonal  elements  only. 

152.  If  A,  B,  C,  -F,  G,  -IT denote  the  minors  of 


a 

h 

9 

h 

b 

f 

9 

f 

c 

show  that 


3S() 


[•KTKKMINAN  rs. 


nx{-  /f//-f-  r/r       /u: }   A//  |-  /:;       ^.rf  f//  \r2 

Ax\-J/f/\  (h     Jlx  I  ////  \'Fz     (;x\i)/-\-i'z 

X  y  2 

II  b(^  lesulvod  into  linear  factors. 
Kind  tlic  factors  in  llio  caso 

a=3,  /;      4,  c  -5,/     1,  y  -  1>,  A  -  3. 

153.  Sliow  tliat  13a.-'  +  5?/  -  10.7-//      2  is  a  factor  of  the 

rcsultantof  llic  elimination  of  r  fioni 

10//  I  13s'-Gyz--L>42 
and  5s''+10a;''-l3a;2-^98. 

154.  Eliminate  y  from 

^*'  (//  f-  y^  1 2' )    ^^y-  (//  I  z)  -f  //V  =  0, 

and  show  that 

ii^  (.j;'  I-  xz  +  2^)  -  -  OTZ  (.r  -|-  z)  +  •x-'s'' 
is  a  factor  of  the  resultant. 

155.  If  a,,  oj  are  the  roots  of  a^x^  +  ^/^i-r  +  c,  —  0, 

)8i,  A    "      "       "      ''    a,x'-\-'HHx-^c.,^^, 
y.,  72    "      "       "      "   LHx'-^'-lh^x-^c,=.^, 
form  the  equations  whose  roots  are 

(i.)2t^i=/?,yH-Ayi,      2w,  =Ay,  +  fty,, 

(ii.)  2?;,    =  a,y,  +  a,y,,      2  ?^,  =  a,y,  +  a.^y2, 

(iii.)  2ifi  =  a,ft+  a2)8„      2'?(;2  =  a,^i+  ajft, 

and  show  that  the  elements  inverse  to  w,  ?;,  and  w, 
respectively  in  *" ' 

1  ^1  Z>2  ^3 

^2     aia./?t;      «2^a      a-^a-^u 
63     aiC^aV     «20^3W       ^3^3 

all  vanish. 


DKTKKMINANTS. 


:W7 


8  a  factor  of  the 


oUi 


•^  ==  0. 


/)  1- :''// 

z)  4-  y^^'  =  ^' 


) 


1^  a;V 


ire 

2  =  aii3i+a2^2' 
rse  to  u,  v,  and  w, 


156. 


\'\'\\o  it'(|uirt'(l  ('(jiiations  arc  tlio  I'lemeiUH  iuvcrau  to 
r,,  r.^,  and  f^  in  thi.s  (lotcriiiiiiaiit.] 

1 1'  (la^  \  h\f  I  rr'  I  ^Ififz  |-  ^lijxz  -\-  2  //.'•//  1)0  roHol va- 
l)lo  into  liiiojir  facioi's,  lliii  coofficioiits  of  y  in  ilioHo 
factons  will  b(i  tlio  roots  of 

and  tlioso  of  z  will  l)n  tlio  I'oots  of 


a?* 


l>y/6  +  r-       0. 


and/  must  hv,  a  root  of 

au'  —  2 /<///«  (-  AV  +  r/^//      ^//;r'  -  0. 

J^xample.    For  what  values  of  i^  will 

x"  -1-  12//  -  -  2l2"'  -  tiyz  -  42-z  f  7.ry 
be  the  ])roduct  of  linear  factors  ? 

157.  A}>ply  problem  155  to  determine  the  conditions  that 

aic^  -[  hif  -}-  c:^  4-  dw^  +  2/yz  -f  2ya;z  -}-  2  A^-y 
-f  2  /.i'7(;  +  2  myw  -f-  2  7?z?(; 

shall  be  the  product  of  linear  factors. 

For  what  value  of  n  will 

x^  +  10?/  -}-  9z'  +  hw''  -Y  LSyz  j  G.rz  -f-  Ga;y 

be  the  product  of  linear  factors  ?    Find  the  factors. 

158.  If  ar^*  +  3  a,a;V  +  3  h^xy''  +  c,/  +  3  a.,x'z  +  3  h,p:z^ 

-\-  c?22'*  +  3  a-^y'^z  -}-  3  ijyz*  +  ^  e:it:y2 

be  resolvable  into  linear  factors,  determine  the  con- 
dition that  these  factors  should  vanish  simultane- 
ously for  values  of  x,  y,  z,  other  than  zero. 

159.  If  u  =  ax'  +  hf  +  cz"  +  2fyz  +  2gxz  -f  2  hxy, 
and  V  =  a^x'^  -f  ^i^^  +  ^i2^  +  '^f\yz  +  2 r/j.rz  -|-  2  Ai^'y, 
find  the  equation  expressing  the   condition   that 


388 


DKTERMINANTS. 


U'}-kv  hIisiH  1)0  the  product  of  linear  factoi-H.     If 
this  equation  have  ecpial  roots  in  k,  hIjow  that  the 

between  w  —  0 


164. 


resui 


y 


and  V  =  0  lias  equal  roots  in  x/z. 

160.  Find  the  square  root  of  the  resultant  of  the  elimina- 

tion of  a  from 

au*  —  2x11^  -\  (I  ---  0, 
au'  —  2i/u  —  a  —  0. 

161.  Eliminate  x  from  ax*  -\- bx -\- c      0  and  r' -   1. 

162.  Eliminate  x  from  ax*-\-bx^-{-  ex'* -{•  dx  +  c  —  0  and 

163.  Eliminate  x  from 


a; 


+  6yla;''-4/ya;+C=0 


and  2if  +  22^  +  x*y  +  3 ^y  +  j5  =  0, 

and  show  how  to  apply  the  resultant  to  obtain  a 
solution  of  the  quartic  in  x. 

Given  /3=Z>  —a,  Pi--  c  ~b,   P-i  —  d—e,    /S^—e—d, 


7=-A-/?.  71- A-A,  y2--=/3,-l3. 


2> 


8- 


7i-y. 

8,-8. 


8, 


72-71. 


show  that 


165.    Show 


a    ^ 

a 
b 
e 

b   e 
c    d 
d  c 

bat  if 

a   b    e 
bed 
c    d  e 

=  0, 

(ac  —  b 

') 

• 
ax 

bjf 

I 

I 

2bxy- 
2cxy- 

then  will 


cy"*    bx*  +  2  cxy  -f  dy"* 


dy"*  cx^-\-2dxy-\-ey'* 


ax 


+  by     bx  + 


n/ 


bx  -\-  cy     ex  -\~  dy 


linear  fjictors.  If 
n  /t,  show  that  the 
'  //  biitween  u  -^  0 

y 
y  ■ 

[ant  of  the  eliinina- 


0  and  ^-^  -    1. 

-{-  dx'{'  e-=0  and 


esultant  to  obtain  a 
a = ft- ft, 


'"•^TKJiMlNANTM. 


3«0 


y   cx''-\-'2.dxy-^ey'    ' 


166.    <'iven.v.^-^4,,.'..|o,,.,-4.,.,.. 
«..'*    f  tt,?;  ~\a.iW        0, 

ft'H-fti-  fyg,V(;        (),' 

«-"«       ft.ft.       y„.y,    ' 
«iiow  thai 


0. 


m-f-H, 


'*.•       « 


I       "i 


/^o    ;«,    /^, 

7"   71   r, 


-17 


a..     II., 


I     </.. 


a 


J  67.      If   /S;„  ~  «"• -f  y8" 


etc.  are  the  roots  of 


'^  y'"  f- S'"  +  etc. 


*l 


in  whic 


ll  u, 


ft  y, «, 


r^a-' 


(lU 


.n    1 


show  (} 


-f- 


n  -a 


lilt 


(-«« 0. 


a-iX 


f'x    (h 


-f I 


'<. 


0. 


3«3   a.. 


0        0 
0 


(t. 


and  tiiat 


^'i«m  a 


m   "m  1    </ 


((,< 


a. 


0 
0 
0 


"»     2     "„,     .1 


C/ 


a 


m 


1X2X3. 


tn 


^%    1        0 


0 
2      0 


Cv 


168.    If  S..  = 


\.  a:„  ,  A' 


1  i 


0 
0 

u 


2   ^' 


^rf^-l-Zy^l, 


7«      3 


A, 


^■"-f  etc.,  then  will 


^0  A,  \^%{a     by 


^%  a;  a;  I = 
^%  *%  >s; 

'%  a:,  ^' 


=   5(<^-<^/(a-^y(i-^^)2 


|;5;5^j.2(.-^n._-,).^. 


G 


^  >%  .v,  ,s^ 

^    ^S;    .S;    ;S: 

^^:^  ^;  ^s,  K 


dy 


if'~^f{h-dy\c-~d) 


eneralize. 


390 


DETERMINANTS. 


169.     If  «!,  €12,  as, 

that 


(In  are  the  roots  of  /(x)"  ~  0,  show 


/W" 


X 

a-i 


X 

a.i 

«3 


«3 


«2 
«3 


<^n-l    ^n-1    a„_i    a„_i 
C  C  C  C 


«1 

«i 

02 

a.2 

«3 

«:, 

a* 

«4 

^' 

«n 

c 

c 

170.    If  /i^„  =  a,"*  +  «r  4-  «3*"  + +  a,r,  show  that 


ii 


1         X 

So  jSy 

^1    ^s; 


^n-1    ^n    ^i 


2»-l 


X 


X 

«1 

o, 

Ox 

a? 

02 

(h 

02 

a: 

171. 


^»-l     ^n    1     0„..i 


Resolve  into  factors, 

0  1      X      x^  :i^ 

1  8q     Si      S2  S-i 

y  S,    S2    S,  S, 

f  S2    S,    S^  S, 

f  S,    S,    S,  S, 

in  which  .i<?,  =  a'^-j-i^  +  c". 


172.    Show  that 


S,    1 

0 

0 

S2  s. 

2 

0 

Ss       S2 

^i 

3 

^m     ^m-1     ^m-2     ^m— 3 


Oo  A3i    ^\-l 


Si  S-i 

s,    s, 

Sn-1    Sn 


O2 
Os 


O, 

03 


a;     o, 
1      1 


0 
0 
0 


?n— 1 

Sy 


Sn 

Sn+l 


of/(.^)"--0,  show 


-1 


«1 

ttl  . 

02 

Ch 

«3 

«3 

a* 

«4 

X 

«n 

c 

C 

"*,  show  that 
aS'.  aS;..,   1 


^. 


2n-2 


«1 

rti   . 

a2 

a2 

«3 

«3 

a: 

«n 

1 

1     1 

0 
0 
0 

•  •  « 

771—1 


I^ETERMINANTS. 


'''>01 


but  =  0,  if  m  >  ,?, 

in  which  ^.^«,.;,^._^^^._^ ^^^^^ 

^  being  any  positive  integer. 


a„,  if  m  = 


w, 


173. 


If  '^'»  =  «i«r  +  «..a.-  +  a3a3-4-a,ar  +  « 


Wo 

w, 


u. 


■5a5  ,  prove  that 


U3 


2        U,        U,        W5f=r0. 


Ui     u^ 


u. 


Ua 


Uo     u. 


u 


6 


IC. 


^5     We     u^     u, 


7        Us       W, 


w„ 


w 


X 


and  ax  -{-  b  ~  y  ^  Q^ 
(See  §  52,  p.  297.) 

and  a;j;  -f-  J  _  „  _  Q 
(See  Ex.  70,  prob.  1,  p.  312.) 

176.   Express  7i;  A,  7,  J,  and  /--277Mn  d.f       ■ 

form,  given  a<!  d»fo  (I,  ^  "^    "»  determinant 

26  Ex  Is   *'r'\**''  proposition.,  stated  in  prob 
-b.  -bx.  68,  and  probs.  14  and  15,  Ex.  70. 

"•   ^^P""^^  ^-  ^-  -^^  ff.  of  the  equation 

'n  terms  of  ^  i;  ^  gr  „f  t^,  ^^^^^^^   ^  +  '^     " 

of  which  n,  n,  r„  r,  are  the  roots. 


392 


DETERMINANTS. 


178.  Express  11^,  In,  Js,  Gs,  and  IJ^  —  21Js\  as  functions 

of  the  differences  of  the  roots  of  the  quartic. 

179.  Express  A^  as  a  function  of  the  differences  of  the 

roots  of  the  cubic. 

180.     If  X  =  \iU  -f  IJLiV, 

y  =.-_  XiiL  -f-  /x-,?;, 
transforms 

ox^  +  2  lixy  -\-  hf 
into  Au'-\-2IIuv-\-  Bv\ 
find  the  value  of 


A,    II 
II,   B 


a,    h 
h,    h 


181.    If 


182.    If 


X  —  XiU  -\-  fliV  +  ViW, 

transforms 

«a;'^  4-  hi/  +  C2"^  +  2/?/2  +  2^^a'2  -f  2  hxy 

into  ^2^'  +  ^y'^  +  Cw'  +  2  i^yw;  +  2  6^ww  -f-  2  IIuv, 

find  the  value  of 

a     h     (/ 
h     b    f 

9    f    (^ 

X  =  (\,y  4-  fi,z)/(\,y  +  fi,z), 
Qix'^  -}-2biX  +  Ci 

and  a^x^  -f  2  b-^x  +  ^^j 

=  ( vl,y'^  +  2  7^,y2  +  a2'0/(A.y  ~  f^.z)\ 
then  will 

^1 2B,  a  0 

0     ^1    27>\  r, 

.4, 2  ^^2  a  0 

0     A,    2B,  C, 


AUG 

-^ 

H  B    F 

Q    F    0 

rti  25x    ^1    0 

X 

A-i  /Ai 

♦. 

0     rti    26,  r. 

^2  M2 

rr,  2b.,    c,    0 

1 

0     a,    2  b,  c. 

DETERMINANTS. 


393 


7//,  as  functions 
he  quartic. 
differences  of  the 


w 


125,    ^1    OixlXi/^ir 
a,    26,  cA    IX.  f^-^l 
l2/^,    c,    0 


183.  If  the  quartic  (a,  6,  c,  fZ,  t')(2',  1)*  ^^  0  be  transformed 

by  the  homographic  transformation 

then  will 

in  which  J/=    A.i   /xi 

[J/ is  called  the  modulus  of  the  transformation.] 

184.  Find  A^/A^  for  the  homographic  transformation  of 

the  cubic  («,  h,  c,  d)(x,  If  =  0. 

185.  If  K={p'-q')/{p'+q'\    f^i-^2pq/(p'  +  q'l 

A,  =:  -  2pq/(y + ./),     /., = c^-^-  '/•0/(i>'^4-  (Z'O. 

X  ---■  Ai?^  +  /^iv,  y  =  A2W  -f- /A-i^^, 

then  will  x"^  +  2/'  =  w^  +  v^ 

[A  transformation  that  changes  a;i^4- ^2^^+ ^'/H- +  ^'»»''' 

into  Ui^ -{- u-i^ -{- u^^  + +  uj  is  termed  an  orthog- 
onal transformation  of  the  nth.  order.] 

186.  Form  an  orthogonal  transformation  of  the  third  order 

and  determine  the  value  of  its  modulus. 
(See  prob.  181  above.) 

187.  Form   an    orthogonal   tran formation   of    the   fourth 

order. 

188.  Show  that  //,  /,  J,  G,  P  ~  27  J'  are  the  same  for 


both  the  quartics 

aox''-\-2hoxy-\-Coy 
aix''-j-2hiX7/-{-Ciy 

a^x^-\-2i^xy-[-a,,y 


a^x^  -\-  2  hyxy  -\-  c^y' 
a-iX"^  -j-  2  h-iXy  +  f^-^y^ 

hiyK^  -f  2  hixy  ~\-  h.{tf 
c^'^-\-2cyxy^c.{y' 


-0. 


0. 


394 


DETERMINANTS. 


189.  Apply  Example  7,  p.  33G,  to  solve  the  cubic 

.r'  +  3/Zr+G'  =  0. 

190.  Form  the  equation  whose  roots  are  the  products  in 

pairs  of  those  of  x^  -\-2^'^^  4"  ^^  + ''  ~  0- 
[ai=)8y,  .".  atti=:a)8y  =  — ?'.  .'.  xi/-^r  =  0;  eliminate  a;.] 

191.  Form  the  equation  whose  roots  are  tlie  products  in 

pairs  of  those  of  x*  -]-px^  +  Q^^  -{-  rx  -}-  s  =  0. 

192.  a,  13,  y  being  the  roots  of  the  cubic  (n,h,c,d)(x,  ly  -  0, 

form  the  equation  whose  roots  are  a)8-f-y,  Py-\-a, 
ya-\-p. 


0. 


193.    If  a,  )8,  y  be  the  roots  of 

a:  0  0  -  a 

I  X  0  b 

Q  \  X  —c 

0  0  \  d 

then  will  ^y,  ay,  a^  be  the  roots  of 


=  0. 


?/  0    0    0     1 

b  ?/  0   1   0 

0  b   1   0   0 

0  abed 

a  b     c     do 


194.    If  («,  b,  c,  d)  (a-,  yy  -  A  {x  +  e,yf  +  B {x  +  e,y)\  show 
that  Ox,  6o  are  the  roots  of 


a 


bed 
10     0' 


=  0. 


195.    If  the  cubic  (a,  b,  c,  d)  (x,  Vf  =  0  be  transformed  into| 
a  cubic  in  y  by  means  of  the  equation 

?/  =  a  {ax  +  b)  -\-p(ax'  -\-?>bx-^  2c), 

show  that  this  cubic  is 


the  cubic 

re  the  products  in 
r  -=  0. 
,.=0;  eliminate  a;.] 

ire  tlie  products  in 
-|-  ra;  +  s  ==  0- 

are  ajS  +  y,  ^Y  +  a, 


)ts  of 


^'  +  5(a:  +  M'.s^^^ 


10  be  transformed  in toj 

[equation 
3fea;  +  2c), 


DKTKKMINANTH. 


395 


y  —  ab-'l(ic  aa     'Sfifj  —fia 

pd  y~ah-\-    pc         —ah 
ad  3ar?  ;       pd  ?/  +  2a/>-|-/Jc 


0. 


196.  J)etermine  the  condition  that  the  roots  of 

(a,6,c,cZ)(a:,  ly'-O 
may  be  formed  from  those  of 

by  adding  the  samo  quantity  to  each. 

197.  Given  (i:x^' -{- 3 bx^ -]- o rx  +  d -^  0 

and//y'^  +  2%  +  Z;-0, 

express  in  the  form  of  a  determinant  equated  to 
zero,  the  equation  whose  roots  ..re  Ix  -\-  my. 

198.  Being  given  the  cubic  (a,  b,  c,  d)(x,  1)'*  =  0,  express 

???,  p,  and  q  in  terms  of  a,  b,  c,  and  d,  so  that  the 
vahies  which  7n9/ -\- 2  2>y -\~  q  takes  when  y  is 
replaced  successively  by  a,  P,  y,  the  three  roots  of 
the  cubic,  are  the  three  roots  in  the  order  p,  y,  a. 

199.  Determine  the  relation  that  must  exist  among  the 

coefficients  of  the  cubic  (a,  b,  c,  d)  (x,  \y  =  0,  in 
order  that 

Aa+Bp+Cy^^O, 

a,  p,  y  being  the  roots  of  the  cubic. 

200.  If  a,  p,  y,  a^,  ^i,  y,,  are  the  roots  of  the  cubics 

(a,b,c,d){x,rf=-0,     {ai,bi,ci,d)(x,iy  =  0, 

form  the  equation  whose  roots  are  aui-f/Sft  +  yyi, 
etc. 

201.  a,  P,  y,  S  being  the  roots  of  a  quartic,  form  the  equa- 

tion whose  I'oots  are 

PyS  +  yS  +  pB  +  Pyi-p  +  y  +  8,  etc. 


396 


DETERMINANTS. 


202.  Also  till!  oqiiutiun  whoso  I'uots  are 

(«-,8)(«-7)(a-S),  (^    y)(;3-S)(/3-u), 

203.  And  the  equation  wliose  roots  are 

(a-;8)(7-8),   (a-y)(8-;8),   (a   -R)(li   -y). 

From  this  result  prove  that  if /''  — 27  J^'^  —  0,  the 
quartic  will  have  equjil  roots. 
(See  prob.  14,  Ex.  70,  p.  312.) 

204.  If  a,  fS,  y,  S  be  the  roots  of  {a,b,  c,  d,c)(x,iy  ~0, 

then  will    the  equation  whose  roots  are  (a  —  fi)'\ 
(a  —  y)'^,  etc.,  be 


—3a 


a^z 


\a'z'-\AIh-\-I  alz^-^J  -2Jz 


a?z 


0. 


205.  a,  p,  y,  8  being  the  roots  of  the  quartic 

(a,  h,  c,  d,  c)  (x,  ly  =  0, 
express  the  product 

\x  -  (13 -yYllx^- (a- Syi 

in  terms  of  a,  b,  c,  d,  and  a  single  root  of  the  re- 
ducing cubic  f  ■—  It-\-  2J=^  0,  and  hence  form  the 
equation  of  the  squares  of  the  differences  of  the 
quartic.  . 

206.  Solve    \x-a{aP-\-y^)\\x~a{ay-{-ph)\\x-a{a^i-Py)\ 
=  1C) 


a 

b     c 

b 

c     d 

c 

d    e 

in  which  a,  ^,  y,  8  are  the  roots  of 

{a,b,c,d,e){x,\y  =  (^- 

207.    Find  the  relation  which  connects  the  coefficients  of 
the  cubics 

U={a,  b,  c,  d)(x,  l)-\ 
V  =  (a\b\c',d'){x,iy, 


y)(^--8)(/J-a), 

-P),    (a   -8)(^      7). 

f  r  -  27  J'  =  0,  the 


!  roots  are  (a  —  P)'\ 


single  root  of  the  re- 
,  and  hence  form  the 
le  differences  of  the 


3ts  the  coefficients  of 


DETEKMINANTS. 


Of  >►» 

tUatA(/  +  ^rmayI)e  a  perfect  cube. 


*  and  (^,  such  tliai 

that  \a     h     c  I  =  0 


6>)^ 


show  that 


d 


209.    If 


and  form  th( 
1 


quadratic  determining  $  and  0. 


1 


1^  +  --+- 
P      y  —  a      a 

in  whicli  «,  ^,  y,  s  ^^,^.  <j 


S 


0. 


(«,  ^,^,r/,e)(.r,  1)«^0, 


10  roots  of 


show  that 


a 

b 

c 

b 

c 

d 

c 

d 

e 

0. 


210.    If 


a 


V/5, 


a. 


and  a+y^  be  three  of 


0. 


the  quartic(«,^,.,,/,,)(^^1^.^0,  show 
0.  Q  n 


0. 


the  roots  of 
that 


^7^  216^,  5(a'I~Sjp) 


3/7; 

0. 


0 
G 
0 


^0. 


27/7 


21(7. 


211.    Show  how  to  solve  the  quartic  (a 
by  assumintr 


5(a'I~S  IP) 


lllU^ 


and  eliminating 


^1.    ?^2,   Vi,   U2 


.'if- 


f 


'M)S 


DKTKUMINANTH. 


I      I      0 


1  u,  t\ 
1  n^  r, 
0  0    0 


r,      r 


A       (•      f        (I 


.();    i.o.,4/-'         //    I    /O.J 


212.  Show  tluit  llio  reduction  of  the  (luiirtio 

to  iho.  hicjiiiiM ratio  Ibnn 

(.r'4-1)''  I-  A.r{,v'  I   1)  |-/?.c'-0 

dopeiuls  upon  the  solution  of  the  cubic 

(ax*  -}-  4  bx^  +  6  ex""  +  4  r/.i;  +  c)  (a:r  f  />)^ 
=  « (<u'='  4-  3  /)x'  -h  3  ex  +  c/)^ 

213.  If  a,  p,  and  a,,  /:?,  are  the  roots  of 


(I 


•eapectively,  show  that 


a 


'■lb 


0      a      2b 


0 


2  b 

0\ 


0 
c 
0 


2/; 


h    e, 


0. 


r'^  I  2bx  -I-  r?  --  0  and  a,.r"^  -f-  2/>,.r  -f  r,  -  0, 


EE  aW{a  -  a.)  (a  -  p,)  (p  -  a.)  (/3  -  A). 

214.  Simikirly,  resolve  the  resultant  of 

ax^-{-  3 bx^-]-^cx -\-d=0  and  ai.r'4- S/^i.-r  +  Ci= 
into  a  product  of  the  difTerences  of  the  roots  of  tlj 
two  equations. 

215.  By  eliminating  g,  A,  k,  and  /  from 


DETKUMINANTH. 


8liU 


0. 


n\J^o.] 


['^ 


iirtio 


tho  cvibic 
a,.T' -1-2610:  4-^1 


-0. 


it  of 

)anda,r'4-26i'^  +  ^^^ 
nces  of  the  roots  of  tl| 


I  from 


|»rov<"  lliiit 


(I     n      (ta 


I   Js 


-  1  aft  \  rd  afi  |  yS 
1  he  I  (l<i  fty  I  ha 
I      ar   I   Ar/     ay    |   /i« 


1 

1     h    li    hii 

1     d 

=  (a     «(/v      r;)(y     8)(./     a) 

(a      A) (/J       y)(6.  --fi)(8     -a) 

=  («       y){o      d){^      fi){h       a) 
{a       c){y       h){d-h){li       a) 

=  (a      8)0/      b){li      y)(.      a) 
(a    -r/)(8      )8)(6    -c)(y       a). 


216.    Similarly,  provu  that 


1 


and  that 


1 


1     a     (la 
1      />     bp 


ry 


:(a     b)(P     y)c    (a     I3)(b-c)a, 


a  \  a      an 


1     6   \  P     hfi 
I     <?    |-  y     cy 

217.    a,  ^,  y,  8  and  ai,  )3i,  yi,  8,  being  the  roots  of  two 
quartics,  prove  that  if  PJ^^      /I'V',  then  also  will 


a       tt]       act; 
1      )8     P,      PP, 

8.     ^h, 


1 


I 


0. 


218.  If  u,  V,  w  denote  the  roots  of  aV  —  a/a: -f  2 »/— 0, 
and  ?^l,  Vx,  tVx  the  roots  of  ai"V  —  a,/,a:  -\-2Ji=  0, 
a,  /8,  y,  8  the  roots,  and  /  and  J  the  invariants,  of 
(a,  6,  c,  (f,  e)  (a:,  1)*  =  0,  and  aj,  )3,,  y,,  81  the  roots,  and 
/i  and  Jy  the  invariants,  of  (ai,  61,  c,,  c/,,  ^i)  (a:,  1)*  — -  0, 
then  will 


400 


a 

DETERMINANTS. 

u 

1 

O]       aai 

-  4 

1 

u, 

1 

/3 

A  m 

1 

V 

Vi 

1 

I 

Vi    yyi 

1 

w 

w^ 

1 

8i      88, 

219.  If  a,  /?,  y,  S  be  the  roots  of  (a,  6,  c,  d,  e)  (x,  l)*  -  0, 

and  tti,  ^,,  y,,  8,  those  of  (a,,  6,,  Cj,  «?,,  c,)(a;,  1)*  —  0, 
form  the  equation  whose  roots  are  the  twelve  dif- 
ferent values  of 

1      a      Qi       aa, 

1  P  Pi  m 

1    y    yi    yyi 

1     8     8,      88, 

220.  If  (a  -  a,)  (/J  -  ft)  +  (a,  -  j3)  (ft  -  a)  -  0, 

in  which  a  and  fi  are  the  roots  of  ax^  -{-2bx-{-c=0, 
and  a,  and  ft  are  the  roots  of  a^x^  -\-2h,x-\-Cx  —  0, 
show  that 

acx  —2hhy-\-  cai  =  0, 
and  that  aci  —  2bbi-{-  ca,  is  a  factor  of  the  invari- 
ant cT  of  the  quartic 


quari 
{ax^  -\-2hx-}-c)  {a^x""  -f  2 b,x  +  c,)  =  0. 


221.    Reduce 


to  the  form 

\x''  +  y''-\-Ax-{-B\''-=ax^-\-bx-\-c,       ^ 
and  show  that  6^-4  .4^^>  +  4  ^V  =  0. 

222.    So  determine  ^  and  I  in  terms  of  a,  b,  and  c,  that 
(a:'^+  y'+  z')'-f  2  aa:^-}-  2  %^+  2  C2'+  2  ^a:+  ^  -  0 

may  for  y  =  0  assume  the  form 

(.!•- +  2' +  w)  (a;' +  r' -1- w) -- 0  ; 

and  for  z  —  0,  the  form 

{x'  +  2/H  wi)  (a;^  +  f  -I-  n,)  -  0. 


6.  c,  rf,  e)  (.r,  ly  -  0. 

,  c„  ^„  ^,)  (^'.  1)*  =  .^' 
are  the  twelve  dif- 


i-a)-0, 


a  factor  of  the  invari- 


ax''-{-bx  +  c, 

of  a,  b,  and  c,  that 

f-i-2cz'-{-2Jcx-\-l=-0 

rm 
n)-0; 


I)ETEKMINANT«. 


401 


223.    Ifa,  ^,  y,  S,  lour  of  til 


e  roots  of  the  quint 


be  connected   by  tlie  rchition  a-f  ^ 
that  €,  the  fifth  root,  will  b 


ic 


y  hS,  show 


z  =  a€-j-  h 


«;  given  by  the  equations 
87/2-f-i(]r/..(). 


224.    If  a,  ^,y,S,  four  of  the  roots 


oft 


th 


(«,  ^.  ^,  d,  cj)  (..  1)5  ^  o_ 
Je  connected  by  the  rehation 

en  will  e,  tlie  fifth  root,  be  det 


ne  qiiintic 


u 


■)0 


a 


ci'inined  I 


ae  +  /6, 


S//2:-i-16  6^:    0. 


in  which  a,y3, 


225.    ^n--my-~^)H^-^y){h~-a)^^ 
y,  S  are  four  of  the  roots  of  the  quinti 

then  will 

in  which  ' 

A  =  aV>V  (/^c- +  m -H  a^) 

226.    Reduce  (a,  5,  .,  d,  ej)  (x,  y^  to  the  form 

and  hence  prove  that  ( )(.,  l)5^o  can   be  re- 
duced to  the  form  / 

^(x+iy~-mx'-n=^0. 


227.    Resolve 


0  111 

1  0  a'  b' 
1  C  0  a'^ 
i  b'     a'     0 


into  linear  factors. 


402 


DKTKRMINANT8. 


228.    a,   (i^    y  ha'ih^    tlie    roots    ol"    x^     px*  ~\-  qx 
exprcHH 


-  r  -  0, 


0 

a 

P   y 

"T" 

0 

1 

1 

1 

y 

0 

y   /^ 

1 

0 

y 

^ 

fi 

y 

0     a 

1 

y 

0 

a 

a 

ft 

a      0 

1 

ft 

a 

0 

ill  terms  oi' p,  q,  and  r. 


229.    Fiiul  tlio  Viiliie  of 


0  a 

a  0 

h  c 

c  h 


h 
c 
0 
a 


p. 
h 
a 
0 


0 
1 
1 
1 
1 


1     1     1 

0     a     /; 
a     0     b 
b     b 
c 


1 

c 

c 

0     a 


c 


a     0 


230.    Show  that 
0 

b' 

3 


c 


a' 
0 

f 
ft' 


b' 
0 


ft'' 
a' 

0 


=  16/  {f  -~  aa)  (p'  -  bft)  (p'  -  cy) 
2p^  =  aa.  -f  b^  -f-  cy. 


231, 


0  1  1 

1  0  a^ 
1  a'^  0 

1  ft'  & 


ft' 

0^ 


a 


1 

t 

a' 
0 


0. 


then  will 


(a'  +  b'  +  ^^  +  a'  4-  ^'  +  /)(aV  +  b'ft'  4-  cY) 
=  2  a'a\a'+  a')  +  2  b'/3Xb''-\-  /8')+2  cV(^'+  f) 


232.    If         a'+)8^  +  a^  =  c' 

i-a'+ya=b' 
then  will 


—1)3^  +  2'^  — r  —  0, 


1  1 

y  P 

0  a 

a  0 


1  1  1 

a  h  c 

0  b  c 

h  0  a 

c  a  0 


bp)  (/  -  cy) 


\\b'+p')-^2cY(c'+y') 
o'a'P'  +  a'b'c\ 


I'KTKUMINANTS. 

lo; 

V      A       1       1       1 
|1      /      fy     a'     0 

i      -0. 
1 

r 

233.    U          .My'      2a^^,>,     „.,i 

z'+x'-   2y:x  ^  b\ 
then  will 

1    y   /? 

Pal 

-  0, 

^      1       1       1       1 
1      0      x'     ,f     z' 
1      x'-     0       6*"^      // 

1      z'     h'     a'     0 

=  0. 

234.    Prove  that 

• 

0           X           y         , 

-^-     0        r      A 
-y       0      0      a 

2     ~-^h     -a     0 

^  (a.  — 

h  -1  «)', 

and  generalize  the  theorem.   ' 
235.    Evaluate 


X 

1-' 

2 

y 

X 

y 

z 

X 

and  use  the  result  to  prove  that 


\     \    y\  a»d  use  the  result  to  prove  that 

y    z    \\ 

+  2(2^--'-l)(2y'-l)(2z^_l) 


404 


DETERMINANTS. 


\ 


■4 


237. 


a         h         c  r/    '  = 

—h       a  —d  c 

—  c       d        a  —  b 

~d  —  c       b  a 


a  -f  ih      c  -\-  id  '^ 
-  c  +  id    a  —  id 


=  (a''  +  62  +  c^  +  (iX  eH  1  =  0. 

Apply  this  identity  to  prove  that  the  product  of  the 
sum  of  four  squares  by  the  sum  of  four  squares  can 
be  reduced  to  the  sum  of  four  squares. 


238.    Prove  that 


240. 


qc 
-qd. 

pqd 
qc 

a 
b 

a 

a  pb 

—  b       a 

—  c  pd 

—  d  ~c 

Hence  prove  that 

(a^-\-2^b''+qc'-\-2^'l(^')(^^+P^'+q^'+2^Q^') 
=  (a A  i-pbB  +  qcC+pqdBf 

+^  (-  aB  +  bA~  qcD  +  qdCf 

+  q  (-  aC-\-pbD  -f  cA  -pdBf 

-j-pq(~aI)-bC+cB-\-dAy. 


239.    Show  that 


a"         2ab        b' 
aa     a/3  +  ab     b/3 


=  (a(3  -f  abf, 


and  generalize. 

If  u  =  (x  —  ai)  (x  —  a.^)  (x  —  a-^) (x  ~  rt,.) 

=  .r"  —  pi^;**  ^  +^2^'""'' {~~)"Pnj  show  that : 

0  1111 

1  X     a^    as     tti 
1     ai    X     a-i     «4 

1        «!        «2       ^         ^4 


«i 


a-,     a. 


X 


=  -  ?i.r"-'  +  {n  —  1)  pi.r"-'  —  {n  -  2)/)2^'"-^  +  ■ 


ih      c  -f-  id 
id     a—  id 


+  1  =  0 

at  the  product  of  the 
m  of  four  squares  can 
L-  squares 

pqd 

''\ 

a 

l^JrpB'-^qC'+pqD') 

pqdBf 

iD  +  qdCf 

'A-pdBf 

?B  +  dA)\ 

=  (ap  +  ab) 


=  w-f  5 


DETERMINANTS. 


=  ^-^)"J'«"-'-(«-%,^-'+ 


■}. 


((rll 


243.    Show  that    I  a -fa;         a:  x 

^  ^        y-^x 

X 


X 

X 
X 


X 


244.    Prove  that  1 0     1     i     i      i  ,      . 


a 


0 (.^-««) 

(— )"p„,  show  that 


and  generah"ze. 

246.    Writing /(.,)  for  (.,-,^v.^^x        ,  '     ' 

that  ^^  '     ""^ fc-^),    show 


f      ^     ^     a    ...     '  «^^ 

o      h     h     c^ 


246.    Show  that    fO 


»-■-'- (n- 2)  p,a;«-H 


a. 


«. 


«3 

^>i     0      a, 

^1      ^2      0       a, 

^1        ^2         ^3         0 

^1    K    h 


a. 


«4      «^ 


A        «5 


=   <^ 


h,      0 


i«.«3a.a.  +  ^,5,«3«,«^  +  ^^^^^^^^^^_^^^^^^^^^^^ 


40G 


DETERMINANTS. 


247. 


248. 


249. 


250. 


a 

0 


h 
b 


a  a 

b  b 

c  c 

a  a 

b  b 


b 
c 
c 
b 
c 
a 
b 


b 
c 
a 
a 
c 
a 
b 


b 
c 


b 
c 


a     a 
b     b 


b 
a 
b 


c 

c 

b 


b 

c 
a 
b 
c 
a 
a 


=  \{a-h){b~c){e-a)r 


h-^+     " 


a      a 


b  '  b 


^:)} 


0  1 
a 
c 


1 
b 

y 

J 


a  a 

b  b 

c  c 

a  a 


1 

b 
c 
c 
b 
c 
a 


1 

b 

c 


1 
b 
c 


a  a 

a  b 

c  b 

a  a 


1 

b 
c 
a 
b 
c 
c 


=  n\(a  ~  b)  (b  --  c)  (c 
X(a'+P+c'-bc 


ca  —  ab). 


Prove  the  two  following   identities,  and    generalize 
them  : 


a~  X        b  c 

b         c  —  X        a 
c  a        b   -  X 


=  (x-So){x'-S,S,), 


Sn  =  a  f  cu"/;  +  (o'^V,     w'  +  o)  +  1  ---  0. 


a—  X      b  c 

b  c  —  X      d 

c  d  a  —  X      b 

d         a         b       c  —  X 


d      =  (x   So)ix-S,)(x'-S,S,)\ 
a 


S„  =  rt  +  i^'b  +  ^■'V  +  i'%     i'  +1=0. 


^DETERMINANTS. 


b  —  c. 


} 


n    \ 

-  ab), 
iiitities, 


and   generalize , 


o>  +  1  =--  0. 


55. 


^•^  +  1  =  0. 


^  Ux^~' \  X  •}- ^  ^,J(^1_\ 
^  x~~2a   / 

^  1  T 


408 


256. 


DETETIAITNANTR. 


ai^a         a2«3      (x—asY 


I  a;— ^a„J 


257. 


0  «!  «2  ^3 

«!  (a;— rti)^       ai^a  «i«3 

«2       «i^2  (a;  — aj^       ^2<^3 

ttg       ai^s  «2«3  (a;  — fts)^ 


=  -  C7a;"-^  2 


a. 


a; 


2  a 


'TO 


258. 


X  —  Ax  ^2  «3 

ai      x  —  Ai      «3 

«!  flj        ^  ~  Ai 


=x{x-8Y-\ 


tti  tti  as      x  —  An 

S=ai-\-ai+ +  «n,        ^m  =  ^— <^TO- 


259.    If  /(a,  i) 


a  -  k  ,  a—  I  [  g  —  '^  I  a  —  n 


b-k '  b-l'  b 
show  that 

f(a,a)  f{b,a)  f{c,  a)  f(d,  a) 
Ka,b)  J\b,b)  f(c,b)  fid,b) 
f(a,  c)  f(b,  c)  fie,  c)  f(d,  c) 
f(a,d)    f{b,d)    f{c4)    f(d,d) 


m     b 
=  0. 


n 


260.    Expand 


261.    Expand 


\  —  c  2a 

2b  \-c 

3b  2b 

2b  36 

a  b 

b  c 

c  d—\X 

o?  +  X        e 


3a  2a 

2a  3a 

X-c  2a 

2b  k-c 

c       d—  X 
d-\-\X      e 


e 

f 


f 

9 


DETERMINANTS. 


262.    Show  that  if 


409 


{^~a){x~b)(x--c)~:^- 


] 


a-j~b 
4b 
6b 
4b 


[a 


a  +  b  — 
4b 
6b 


ta 
4a 
a-\-b~ 
4b 


■px'-j-qx-r, 


a 


6 
4a 
a-i-b- 


P 


0-16 
0       0 


r     0 
P        q    r 


263.    If  a-{.b-\.c-^0,  ab-\.bc-]~ 


P    q 

'  P 


ca 


an 
-A 


A^  =  8 


—  m  ,a_ 

—  m     b 


=x(x-sy-\ 


~q,  abc=--r,  then  will 


264.   Show  that 


A^  (/  -  2r=')  -f  g^ 


—  a„ 


n 


■2bp-~2 


^y      cip  +  a^ 


aP  +  ab        A  -  2 


«y-f-ac 


w 


~2^y  by-\-pc 

h-V^c        \~~2aa~-2b(3. 


,  a) 
\c) 


=  0. 


265.   Show  that 


^). 


'a' 


Ca-{-c 


J^t 


h^ 


2a 

3a 

2a 

\-c 

d-X 

e 

f 
9 


J.  J 


A  +  aa  +  a'a'       ba  +  b 

erminant,  and 


wherein  n  is  the  order  of  the  det 

A  ==  aa' -i- bb' -j- cc' -i- 


B 
C 
L 


««'  -f  A?y5'+  yy'+ 


410 


DKTERMINANTS. 


266.     SliDW   lluit 


Ida     a^ 

~ 

1   a 

X 

\    (i  P    P' 

1   ft 

1     y     y'    yy' 
1     8      8'     ?>^' 

1       1       1 

a  1/3  y   I    8'  8  ly 
a(i         yS'         8y' 


ami  generalize  the  proposition. 
267.    Show  tluit 


h 


h. 


0 
0 


and  generalize  the  proposition. 


(I. 
0 

0 


a,i  />4 1  X  I  ''i  (li 


d}^ 


268.    Given  (.r,  --  a:,)^  +  (y,  -y,)'^ 

(.r.-.r;,)M-(?^-y3)■^-^'^ 


1 

1 

.r, 

yi 

0 

1 

X-i 

?A 

1 

1 

.T3 

:'A 

0 

1 

x^ 

y4 

then  will  16  B''  =-  4  A'^^-'^  -  (a*^  -  l?  +  c^-  J'y. 
If  Ilk  ~  ac  +  ^c?  and  2  s  —  a-^-h  -\-  c  -f-  ^^, 
then  will  B''  =~-  (s  -  a)  {s  -  Z*)  (s  --  c)  (s  -  d). 

-0 


rtl 

h 

i'l 

.Ti 

Vx 

^1 

rt.2 

L 

C'l 

X., 

Ih 

22 

^3 

b. 

C-i 

X., 

y« 

^u 

269. 


is  satisfied  by  any  one  of  twelve  systems  of  three 
equations  each  ;  find  them. 

270.    Given  x^'  -f  y;'  -\-  z^  --  1,       XyX^^  +  3/13/,  +  z,2.,  =  0, 
x-i  +  Vt  -I:  z.,'  --•  1 ,       x,,x.,  -\-  y,,y-,  -f  ^,2-3  ---  0, 


DKTKHMINANTH. 


1  1  1 

a\/3  y  I   8'  8-f-y 
afi         yS'         8y' 


n. 


X  I  r,  (I,  I 


n. 


2  „  ^2  _|_  ^2_  ^^2y2^ 

b)(s-c){s-d). 


271. 


272. 


273. 


411 


prove  (hat 

^'.^^•./-f-.V==l, 

mid  ir 


■^2 


and     .r,  -  ^ 


r  tlicri  will  A      ±  1 


.'A  -  A 

y. 


^x 


"•i 


-.•I 


A 


0. 


X 

c 
c 

d 
d 

V 


a 

X 

c 
d 

y 
h 


a 
a 

X 

y 
b 

b 


h 
b 

y 

X 

((, 
a 


b 

y 
d 

c 

X 

a 


'^■\~y  ci  \  b  a-{-b 
o+d  x^-y  a-\-b 
^+d   c-\~d   c+d 


X 

c 
c 

d 
d 

y 


„,  2 


a 

X 

c 

y 
d 

y 
b 


a 

a 

X 

<i 

y 
b 

b 


X 


X., 


welve  systems  of  three 

^2  +  yiy-i  +  ^1^2  =  0, 
r,  +  ?/2?/3  -f-  222:3  ^  0, 
Vi  +  ?/:,?/!  +  2-32:1  ==  0, 


^^y-i 


X. 


V 

V 

V 
2r 

V 
V 
V 


yi 
y^ 
y-^ 


b 
b 

y 

H 

X 

it 

it 


b 

y 
d 

'I 

c 

X 

a 


y^^\x~y 


b 


a~b  a-b 
c  -d  x-y  a  ^b 
c  —  d   c  -  d   x~y 


X  h- 


P 


P 


1  x-\-y  a  +  b  a  +  b 
(J  c-Yd  x^y  a  +  b 
<1    o  +  d    a+d   x-l-y^ 


\   ^*yi  yi' 


x^ 


y^' 
yi 


yx^x 
y-zz, 
yiiZ.i 
y^^i 

y^z^ 


r,    2 


Zf: 


pf&  |.|::ia  S:?rftf 

1^^,-z.Y,  ^„x,~x.,z,  x.,r'yt.k 


412 


DETERMINANTS. 


Y       = 


^m,  n 


Z     = 

•*■  m.  ti  — 


m,  n 


-^m,  n  = 


274.    If  a,  i,  c,  c?,  e,/  denote  the  six  determinants  that  can 
be  formed  from  the  array 


{ 


a,  ^,  y»  S, 


then  will  ad  ■}- be -\-  cf  =  0. 

275.  Prove  that 

I      -         —  I  aiC?,6?3/4 i^  I  I  ^1^,6?^, in  I 

i .  =1  aAe^fi tn  I  I  Cid^eji t^  |. 

276.  If  An,  Bn,  On  are  the  inverse  elements  of  |  ai^a^a  |  with 

respect  to  a„,  b^,  c^,  show  that 


1     1     1 

Ci       (?2       ^3 
Uj      Cg      C3 


277.    If  I  ^1  ^2  C'a  I  be  the  reciprocal  of  |  ai  b.^  c^  |,  and 


1 

1 

1 

A, 

a., 
A, 

«3 

A, 

1      1      1 

+ 

^1     ^2     ^3 

^1   J^j  ^3 

«! 
^ 

a^   as 
^2    ^3 

^2      ^2     C'i. 

= 

^1 

^2     ^3 

ag      ^>3     C3 

u 

Wi 

Vi 

IVi 

V 

Ui 

Vl 

Ul 

w 

278.    If 


element  for  element,  and  U,  V,  W  be  the  principal 
diagonal  elements  of  |  Ai  B^  C^  p,  then  will  ^ 
uU-{-vV+wW 

~  3 1 «!  ^2^3 1^+  2  uui-{-  2  vvi^-\-  2  ww^—  6  u^ViWi, 

-aA-^bB-^  cC+  dD, 


a 

b 

c 

d 

b 

a 

d 

c 

c 

d 

a 

b 

d 

c 

b 

a 

prove  that 


1 


eterminants  that  can 


'v* « I 

merits  of  |  aJ}iC^  \  with 


1 

B, 


+ 


1    1     1 

Ci     C2     Ca 
Cj    02    C3 


of  I  «!  &2  Ci  I,  and 


w 

Wi 

Vl 

■w;! 

V 

Wi 

^1 

Wi 

ii; 

F,  ir  be  the  principal 
%  \\  then  will 


DETERMINANTS. 


279.    If 


B  A     D    C 

^^  ^    A     B 

B)  C    B 

U  X  y 

Z  U    X 


y  8   a 


^  =\z    U  X 

y  z  u 

X  y  z 
prove  that 

Uab\      haV 
\c  d  a'       '      - 
in  which 

«=|«  b  c 
dab 
c  d  a 

with  similar  expressions  for  ^ff,  y,  §. 

280.    Given  x^  ~vz~~r,      .2 

yz~-a,    y^^zx^b 
*  show  that 

\c  a  b  \ 
'  \b  c   a\ 


z^~xy^c. 


^81.    If 


yz  —  ii"  ^  ^^2^ 


vw~xu^d\    wu 
prove  that 


2^^  —  v^  =r  b^ 


■yv 


xy 
uv 


c\ 


zw  =f\ 


and  solve  the  equations. 


414 


DETERMINANTS. 


282.  It'  I  Ao  Bi  r; A\  I  bo  the  reciprocal  of  |  a°  i  ci^ /;**!, 

prove  tluit  {~iy "Ar/An  —  sum  of  all  the  products 
of  />,  c,  (I,  k,  taken  n  -7'  at  a  time. 

283.  Evaluate 


L 

M 

N 

• 
• 

1 

1 

1 

I 

7)1 

n 

/ 

m 

n 

P 

771^ 

n' 

P 

in^ 

n 

in  which 

M={a,h,c,d){m,  l)^ 
N  =  (a,  b,  c,  d)  (w,  ly. 


284.    Prove  that 


x^ 


a;  0  0 

0  2/  0 

0  0  z 

0  z  y   2i/iZi 


X., 


y-i 


X\X'i 

y^y-z 

Z\Z2  z^ 

Z     0    X    2  ZiXi     ZxX'i  -f  Z^iXy    2  z-p^a 

y   a;  0   2:i:,yi   a;,y2  4-a;,yi   2a;,2/, 


=  I  a;  ?/i  22 


285.    Prove  that 

a;   0  0 

0  2/  0 

0   0  0 


x,^ 
2/1^ 


5/2 

2/2' 


I 

m 
n 


0  z  y  2y^Zx  2y.,z^  2p 
z  Q  X  2ziXi  2Z2X2  2q 
y   X  0   2xi7ji    2x.,y.,    2r 

-\-p{Y,Z,-^Y,Z,) 
-^q{Z,X,-\-Z,X,) 


in  which 
A  = 


y  yx  2/2 

Z      Z\     Zi 


Xy  is  the  minor  oixx  in  A,  et( 


1 


DETERMINANTS. 


415 


.rocalof  |a°^^6-^ ^"1, 

am  of  all  the  products 


286.    Evaluate 

A     J/    G    JV    X  I. 

a     F     (^    L 
N    M    L    1)    xo 
^      y      z      XV     0 

in  whicl,  A   II  C  A  etc.,  are  the  complements  of 
«,  o,  c,  d,  etc.,  in 

^^     ^'      !7    n  \. 
h     b       f    yo  I 

V     f     ^c     I 
fi     m     /     d 


287.    If  I  a,  b,  c,  I  :=  0,  then  will 

y  <^2  h\\y  c,  «., 
^  ^-s  f^A\z  c,  a. 


X  b,  c, 

y  K  c, 


be  a  square. 


288.    If 


"    ^    ^    c/    0   0  1  =  0,  then 
^h  bi   c,  di  0   0  ' 

(h  K  c,  d^  0  0 

^  a  b    0  c  d 

0  «i  <?'i  0  ci  dx 

0  a.,  b,0  c,  d,  I 

«    c    d  \~\a    b    d 


will 


«i  <?i  <f, 

«2     ^Q     C/o 


«i  bi  di 
(h  h-i  d-i 


b    c    d 

bx  Cy  r/, 

h  c.^  d., 

0. 


289.    Show  that 


^     b    c    d 
ct     c    d  e 

2b    d  e    f 
\ic    e   f   g 


+ 


5  the  minorofajiin  A,et( 


c    2c    e  f 

d    Sd  /    g 

-h\a  b  c     0  !  =  0. 
bed   d 
c  d  e   2e 
d  ef  3/j 


f     0     0   d\-]-\a  b    0    d 
b     b     del       b 


c  c  e 
\c  d  2d f 
\d  e   Se  g 


410 


DETERMINANTS. 


290.  Given  Ai  =  \  liVi^p^Y  -^  \\ ^■irth\  \  f^^^h I  I h^h  \  \ , 

A-i  =  I  liTn^p^  I'  -i-  S I  liiUi  I  I  hint  I  I  /tin ,  |  j , 

udj  =  I  liVL^Pi  I '  -i-  j  I  /,7Ua  I  I  liHh  I  I  /4W< ,  1 1 , 

find  the  value  of  Aili  +  ^./j  -]-  A^l^  +  ^4^4. 

291.  Gi\en  Uii=aiX-\-bii/-{-CiZ,       Un^^h-^x -\- a-iy -\- dxZ, 

Un  =  ^i^^'  +  Ky  4-  c,2,       w,3  =  e-iX  -h  %  +  d.,z, 

W2I  ^^  Wl3»        ^31  ^^^  "^13)        ^32  ^^  ^^23> 

W,   =  XUn  +  2/W12  +  2Wl3, 
W2  =  arwai  +  2/W22  +  ZWw, 

W3  =  xuix  +  ywgj  H-  2W33, 

U=XUi   H-yWa   +  ZW3, 

then  will 


292. 


Uii 

W12    Ui 

U21 

1*22    "^2 

Ui 

Wa     0 

Uii,  11-22,  W33  I  2 

Generalize. 


ai     a2  +  «o     «3  «4  «5 

tta     053  4- «i     a4  +  «o     «6  tta 

tts    a4  +  «2    as  +  ai     «6  +  oto  «7 

^4       «5+«3       «6-i-«2        «7  +  «l  «0 

as  aa 


+  6  C/'|wii  W22I. 


as      tte  +  ^4      ^7  +  «3 

c«6    a7  +  as  a4 

ai  ae  ttj 

is  divisible  by 
a2  -  -  ao     as 


as 

a« 

a^ 

«« 

a, 

0 

a^ 

0 

0 

0 

0 

0 

0 

0 

0 

ao 

0 

0 

tti 

ao 

0 

aa 

ai 

ao 

«i 


a3  —  «!     a4  —  ao    a^ 

a^  —  ^2     as  —  ai     ag  —  ao 

as  —  ttg     ae  —  aa    a7 


ag 

«7 


^4     av  —  as 


as 


—  a4 


ai 

aa 
as 


ao 
ai 
aa 


ae 
a; 
0 
0 

—  ao 

—  ax 


«7 

0 
0 
0 
0 
a© 


Generalize  the  result  obtained. 


I^KTEKMINANTS. 


f^vu 

hm,\\, 

ktih 

l,m,     , 

lit/h 

l^mx     , 

Ims 

|4w,   i, 

\-A^k^r^U 

hi- 

h.^  +  a-^ 

+  c/,z, 

ti  = 

e-iX  +  ^i2/  +  ^22. 

^33  = 

c^x  -\-  d^i]) 

-\-a^z, 

=  l< 

^23» 

417 


^22    "^2 


+  6?7|wii  ^22!- 


^4  «5 

as  «« 

as  a? 

0^7  0 

ao  0 

ax  a-o 

a-i  Qi 


^0 

%i 

^3 


ai 

—  ao 

—  tti 
-a^ 


as 

0 
0 

—  ao 


a, 

0 

0 

0 

0 

ao 


ag    07 
a7    0 
0     0 

1 

0     0 

■ 

0     0 

■ 

0     0 

1 

ao    0 

■ 

«!       tXo 

1 

293.    Sim])lify 

yh.  ~  a,/      a,,a,,  ~  a,,a,, a,,n,,     a.„a.,  I 

na23-a,2r^,,    a..a,, -- a,,"^    «na,,~a   ., 

a„a2,  -  a,,^.,    «.,,,,,  -_  «„^,^ ^^^^^^  __  ^^»^^|« 

in  which  a,,,-a,^. 

294.    If  v(a,^,6')2„.H=    a  A  0  0  .....  0  0  0 

G  a  h  0  0  0  0 

^  c  a  h  0  0  0 

^  ^  c  a  0  0  0 

^000  :::::  ;;■  ;  i 

then  will  '    '    '    '     '^    ^     - 

V  («,  A,  e),„+,  =  0! V  (a»  _  2  ic,  i',  e')^. 

295.  Also     V(^,l,l),.  =  v(.r"-2,l,l)„+v(.-'--2,l,l),_, 

296.  l(J{,n,n,i.)^  1     1     1      1    ,  prove  that 

a"»  /?'«  y«  S« 
a"  ;8'*  y«  5« 
a"    )S^    y^     ^P 

297.    IfA^|«,^,^,.....^,,,aA(}/;^  .denote  the 

result  of  replacing  the  /th,  mi\  ..'..TcolumnH  of  A 
^  ""  ^' '  "«'  Z^-' respectively,  show  that 


A-^X  A<^V'  , 

^  ^i, )' 


the  determinant  on  the  left  of  "  =  "  being  of  the 
order  r. 


418 


DETERMINANTS. 


298. 


!!^ 


1111 

0  111 
0  0  11 
0  0  0  1 
10  0  0 


10  0  0 
110  0 


1110 

1111 
1111 

110  0  0  111 
1110  0  0  11 


=  5. 


11110001 

Show  that  the  value  of  a  determinant  formed  like  the 
above,  but  with  m  units  and  n  zeros  in  each  line,  is 
m  if  w  be  prime  to  n,  but  is  zero  if  m  be  not  prime 
to  n. 

299.  If  {a,h,cJ,g,h){e<i>,e^<i>,iy=^0, 

and  {a,  b,  cj,  g,  h){<l>x,  <^+x>  1)'  =  0» 
find  (a,  ft  y,  K,  A,  ,x)(Ox,  ^+X>  1)'  =  0, 
and  show  that  if 

a  =  a,  P=b,  y  =  c,  k=J,  \==fjf,  fi  =  h, 
then  will 

ac-{-b''  +  2hg-ifh  =  0. 

300.  The    minors  of  order  2n  —  l   of  a  skew  symmetric 

determinant  of  order  2n  are  divisible  by  the  square 
root  of  the  determinant. 


-'  i-\\'^  DEp^;; 


63 


4784   6 


). 


nant  formed  like  the 

zeros  in 

each  line,  is 

3ro  if  7)X 

be  not  prime 

Y     0, 

)'  -  0, 

y-0, 

',  ^-i7, 

\L-h, 

a  skew  symmetric 
irisible  by  the  square 


I 


